408 lines
17 KiB
TypeScript
408 lines
17 KiB
TypeScript
import React, { useRef, useState, useEffect } from "react";
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import { ArrowDown, Check, BookOpen, Grid, RefreshCw } from "lucide-react";
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import SystemVisualizerWidget from "../../../components/lessons/SystemVisualizerWidget";
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import Quiz from "../../../components/lessons/Quiz";
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import { SYSTEMS_QUIZ_DATA } from "../../../utils/constants";
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import { Frac } from "../../../components/Math";
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interface LessonProps {
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onFinish?: () => void;
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}
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const SystemsEquationsLesson: React.FC<LessonProps> = ({ onFinish }) => {
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const [activeSection, setActiveSection] = useState(0);
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const sectionsRef = useRef<(HTMLElement | null)[]>([]);
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const scrollToSection = (index: number) => {
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setActiveSection(index);
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sectionsRef.current[index]?.scrollIntoView({
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behavior: "smooth",
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block: "start",
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});
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};
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useEffect(() => {
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const observer = new IntersectionObserver(
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(entries) => {
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entries.forEach((entry) => {
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if (entry.isIntersecting) {
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const index = sectionsRef.current.indexOf(
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entry.target as HTMLElement,
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);
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if (index !== -1) setActiveSection(index);
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}
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});
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},
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{ rootMargin: "-20% 0px -60% 0px" },
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);
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sectionsRef.current.forEach((section) => {
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if (section) observer.observe(section);
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});
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return () => observer.disconnect();
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}, []);
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const SectionMarker = ({
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index,
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title,
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icon: Icon,
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}: {
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index: number;
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title: string;
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icon: any;
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}) => {
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const isActive = activeSection === index;
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const isPast = activeSection > index;
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return (
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<button
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onClick={() => scrollToSection(index)}
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className={`flex items-center gap-3 p-3 w-full rounded-lg transition-all ${isActive ? "bg-white shadow-md border border-blue-100" : "hover:bg-slate-100"}`}
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>
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<div
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className={`w-8 h-8 rounded-full flex items-center justify-center shrink-0 ${isActive ? "bg-blue-600 text-white" : isPast ? "bg-blue-400 text-white" : "bg-slate-200 text-slate-500"}`}
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>
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{isPast ? (
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<Check className="w-4 h-4" />
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) : (
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<Icon className="w-4 h-4" />
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)}
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</div>
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<div className="text-left">
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<p
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className={`text-sm font-bold ${isActive ? "text-blue-900" : "text-slate-600"}`}
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>
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{title}
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</p>
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</div>
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</button>
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);
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};
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return (
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<div className="flex flex-col lg:flex-row min-h-screen">
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<aside className="w-full lg:w-64 lg:fixed lg:top-20 lg:bottom-0 lg:overflow-y-auto p-4 border-r border-slate-200 bg-slate-50 lg:bg-transparent z-0 hidden lg:block">
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<nav className="space-y-2">
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<SectionMarker index={0} title="Number of Solutions" icon={Grid} />
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<SectionMarker index={1} title="Solving Methods" icon={RefreshCw} />
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<SectionMarker index={2} title="Practice" icon={BookOpen} />
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</nav>
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</aside>
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<div className="flex-1 lg:ml-64 md:p-12 max-w-full mx-auto">
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{/* Section 1: Number of Solutions */}
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<section
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ref={(el) => {
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sectionsRef.current[0] = el;
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}}
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className="min-h-screen flex flex-col justify-center mb-24 pt-20 lg:pt-0"
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>
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<h2 className="text-4xl font-extrabold text-slate-900 mb-6">
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Systems of Equations: Number of Solutions
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</h2>
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<div className="prose prose-slate text-lg text-slate-600 mb-8">
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<p>
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A system of two linear equations represents two lines on a graph.
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The <strong>number of solutions</strong> tells you how those lines
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relate geometrically and algebraically. Every SAT test includes at
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least one question asking you to identify how many solutions a
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system has, or to find a constant that produces a specific
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outcome.
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</p>
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</div>
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<div className="bg-blue-50 border border-blue-200 rounded-2xl p-6 mb-8 space-y-4">
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<h3 className="text-lg font-bold text-blue-900">
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The Three Possible Outcomes
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</h3>
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<div className="overflow-x-auto">
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<table className="w-full text-sm border-collapse">
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<thead>
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<tr className="bg-blue-900 text-white">
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<th className="p-3 text-left rounded-tl-lg">Outcome</th>
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<th className="p-3 text-left">Geometric Picture</th>
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<th className="p-3 text-left">Algebraic Condition</th>
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<th className="p-3 text-left rounded-tr-lg">Example</th>
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</tr>
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</thead>
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<tbody className="divide-y divide-blue-100">
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<tr className="bg-blue-100">
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<td className="p-3 font-bold text-blue-900">
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One Solution
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</td>
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<td className="p-3 text-slate-700">
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Lines intersect at exactly one point
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</td>
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<td className="p-3 text-slate-700">Different slopes</td>
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<td className="p-3 font-mono text-xs text-slate-700">
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y = 2x + 1<br />y = −x + 4
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</td>
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</tr>
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<tr className="bg-red-50">
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<td className="p-3 font-bold text-red-900">No Solution</td>
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<td className="p-3 text-slate-700">
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Lines are parallel — never meet
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</td>
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<td className="p-3 text-slate-700">
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Same slope, different y-intercept
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</td>
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<td className="p-3 font-mono text-xs text-slate-700">
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y = 2x + 1<br />y = 2x + 5
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</td>
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</tr>
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<tr className="bg-emerald-50">
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<td className="p-3 font-bold text-emerald-900">
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Infinite Solutions
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</td>
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<td className="p-3 text-slate-700">
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Same line — perfectly overlap
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</td>
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<td className="p-3 text-slate-700">
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Same slope AND same y-intercept (equations are multiples)
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</td>
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<td className="p-3 font-mono text-xs text-slate-700">
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y = 2x + 1<br />
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2y = 4x + 2
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</td>
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</tr>
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</tbody>
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</table>
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</div>
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{/* Finding k for specific number of solutions */}
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<div className="bg-white rounded-xl p-5 border border-blue-100">
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<p className="font-bold text-blue-800 mb-3">
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SAT Technique: Finding k for a Specific Number of Solutions
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</p>
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<div className="space-y-4">
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<div className="bg-blue-50 rounded-lg p-4 text-sm">
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<p className="font-semibold text-blue-800 mb-2">
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Example: For what value of k does 2x + ky = 6 and 4x + 2y =
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12 have infinite solutions?
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</p>
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<div className="font-mono space-y-1 text-slate-700">
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<p>
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For infinite solutions, equations must be proportional.
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</p>
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<p>
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Ratio of x-coefficients: <Frac n="4" d="2" /> = 2
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</p>
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<p>
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So all coefficients must scale by 2: k must satisfy{" "}
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<Frac n="2k" d="2" /> = 2, so k = 2.
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</p>
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<p>
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Check constants: <Frac n="12" d="6" /> = 2 ✓
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</p>
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<p className="text-blue-700 font-bold">
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k = 2 → infinite solutions
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</p>
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</div>
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</div>
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<div className="bg-red-50 rounded-lg p-4 text-sm">
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<p className="font-semibold text-red-800 mb-2">
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Example: For what value of k does 3x + 2y = 8 and kx + 4y =
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5 have no solution?
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</p>
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<div className="font-mono space-y-1 text-slate-700">
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<p>No solution → same slope, different intercept.</p>
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<p>
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Same slope means coefficient ratios match for x and y:{" "}
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<Frac n="k" d="3" /> = <Frac n="4" d="2" /> = 2
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</p>
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<p>
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So k = 6. Check constants: <Frac n="5" d="8" /> ≠ 2 ✓
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(different, confirming no solution)
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</p>
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<p className="text-red-700 font-bold">
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k = 6 → no solution
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</p>
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</div>
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</div>
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</div>
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</div>
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<div className="bg-red-50 border border-red-200 rounded-xl p-4 text-sm">
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<p className="font-bold text-red-800 mb-1">
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Critical Distinction: No Solution vs. Infinite Solutions
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</p>
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<p className="text-slate-700">
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<strong>No solution</strong>: coefficients are proportional but
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constants are NOT. (Same slope, different lines.)
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<br />
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<strong>Infinite solutions</strong>: coefficients AND constants
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are proportional. (Same line, just written differently.) Divide
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one equation by the other and everything must cancel cleanly.
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</p>
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</div>
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</div>
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<SystemVisualizerWidget />
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<button
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onClick={() => scrollToSection(1)}
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className="mt-12 group flex items-center text-blue-600 font-bold hover:text-blue-800 transition-colors"
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>
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Next: Solving Methods{" "}
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<ArrowDown className="ml-2 w-5 h-5 group-hover:translate-y-1 transition-transform" />
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</button>
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</section>
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{/* Section 2: Solving Methods */}
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<section
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ref={(el) => {
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sectionsRef.current[1] = el;
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}}
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className="min-h-screen flex flex-col justify-center mb-24"
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>
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<h2 className="text-4xl font-extrabold text-slate-900 mb-6">
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Solving Methods
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</h2>
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<div className="prose prose-slate text-lg text-slate-600 mb-6">
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<p>
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The SAT presents systems in many formats. Choose your method based
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on what form the equations are already in — don't waste time
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converting unless necessary.
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</p>
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</div>
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<div className="bg-blue-50 border border-blue-200 rounded-2xl p-6 mb-8 space-y-5">
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<h3 className="text-lg font-bold text-blue-900">
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Method 1: Substitution
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</h3>
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<div className="bg-white rounded-xl p-5 border border-blue-100">
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<p className="text-slate-600 text-sm mb-3">
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<strong>Best when:</strong> one variable is already isolated
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(e.g., y = 2x − 5) or easy to isolate.
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</p>
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<p className="text-slate-600 text-sm mb-3">
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<strong>Process:</strong> Plug one equation into the other to
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create a single-variable equation.
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</p>
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<div className="bg-blue-50 rounded-lg p-4 font-mono text-sm space-y-1 text-slate-700">
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<p>Given: y = 2x − 5 and x + y = 7</p>
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<p>Substitute y = 2x − 5 into x + y = 7:</p>
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<p>x + (2x − 5) = 7</p>
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<p>3x = 12 → x = 4</p>
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<p>y = 2(4) − 5 = 3</p>
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<p className="text-blue-700 font-bold">Solution: (4, 3)</p>
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</div>
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</div>
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<h3 className="text-lg font-bold text-blue-900">
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Method 2: Elimination (Addition/Subtraction)
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</h3>
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<div className="bg-white rounded-xl p-5 border border-blue-100">
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<p className="text-slate-600 text-sm mb-3">
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<strong>Best when:</strong> equations are in standard form (Ax +
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By = C) and coefficients are easy to match.
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</p>
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<p className="text-slate-600 text-sm mb-3">
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<strong>Process:</strong> Multiply one or both equations so one
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variable's coefficients are equal and opposite, then add the
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equations.
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</p>
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<div className="bg-blue-50 rounded-lg p-4 font-mono text-sm space-y-1 text-slate-700">
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<p>Given: 2x + y = 10 and 2x − y = 2</p>
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<p>Add equations (y terms cancel):</p>
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<p>4x = 12 → x = 3</p>
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<p>Back-substitute: 2(3) + y = 10 → y = 4</p>
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<p className="text-blue-700 font-bold">Solution: (3, 4)</p>
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</div>
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<div className="mt-3 bg-blue-50 rounded-lg p-4 font-mono text-sm space-y-1 text-slate-700">
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<p>Given: 3x + 2y = 16 and x + y = 7</p>
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<p>Multiply the second by 2: 2x + 2y = 14</p>
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<p>Subtract: (3x + 2y) − (2x + 2y) = 16 − 14</p>
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<p>x = 2, then y = 5</p>
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<p className="text-blue-700 font-bold">Solution: (2, 5)</p>
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</div>
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</div>
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{/* SAT Speed Tip */}
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<div className="bg-sky-50 border border-sky-200 rounded-xl p-5">
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<p className="font-bold text-sky-900 mb-2">
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SAT Speed Strategy: Sum/Difference Shortcut
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</p>
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<p className="text-slate-700 text-sm mb-3">
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If the SAT asks for a <em>combination</em> like x + y, or 2x −
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y, you can often get it directly by adding or subtracting the
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equations — without finding x and y individually.
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</p>
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<div className="bg-white rounded-lg p-3 font-mono text-sm space-y-1 text-slate-700">
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<p>Given: 3x + 2y = 14 and x + y = 6. Find 2x + y.</p>
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<p>Subtract eq2 from eq1: (3x + 2y) − (x + y) = 14 − 6</p>
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<p className="text-sky-700 font-bold">
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2x + y = 8 ← answer directly!
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</p>
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</div>
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</div>
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{/* Word Problem Translation */}
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<div className="bg-white rounded-xl p-5 border border-blue-100">
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<p className="font-bold text-blue-800 mb-3">Word Problem Setup</p>
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<p className="text-slate-600 text-sm mb-3">
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Most SAT system word problems follow this template: define two
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variables, write two equations (one for each constraint), solve.
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</p>
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<div className="bg-blue-50 rounded-lg p-4 text-sm">
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<p className="text-slate-700 mb-2 italic">
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"A store sells pens for $2 and notebooks for $5. A customer
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buys 8 items total and spends $28. How many of each did they
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buy?"
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</p>
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<div className="font-mono space-y-1 text-slate-700">
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<p>Let p = pens, n = notebooks</p>
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<p>p + n = 8 (total items)</p>
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<p>2p + 5n = 28 (total cost)</p>
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<p>From first: p = 8 − n. Substitute: 2(8 − n) + 5n = 28</p>
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<p>
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16 − 2n + 5n = 28 → 3n = 12 →{" "}
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<strong className="text-blue-700">
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n = 4 notebooks, p = 4 pens
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</strong>
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</p>
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</div>
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</div>
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</div>
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</div>
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<button
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onClick={() => scrollToSection(2)}
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className="mt-12 group flex items-center text-blue-600 font-bold hover:text-blue-800 transition-colors"
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>
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Next: Practice Quiz{" "}
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<ArrowDown className="ml-2 w-5 h-5 group-hover:translate-y-1 transition-transform" />
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</button>
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</section>
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{/* Section 3: Quiz */}
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<section
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ref={(el) => {
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sectionsRef.current[2] = el;
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}}
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className="min-h-screen flex flex-col justify-center"
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>
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<h2 className="text-4xl font-extrabold text-slate-900 mb-8">
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Practice Time
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</h2>
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{SYSTEMS_QUIZ_DATA.map((quiz, idx) => (
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<div key={quiz.id} className="mb-12">
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<Quiz data={quiz} />
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</div>
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))}
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<div className="p-8 bg-blue-900 rounded-2xl text-white text-center mt-12">
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<h3 className="text-2xl font-bold mb-4">Topic Mastered!</h3>
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<button
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onClick={onFinish}
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className="px-6 py-3 bg-white text-blue-900 font-bold rounded-full hover:bg-blue-50 transition-colors"
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>
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Finish Lesson ✓
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</button>
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</div>
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</section>
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</div>
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</div>
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);
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};
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export default SystemsEquationsLesson;
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