System of Two Linear Equations – Algebraic Methods

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Key Formula - General Form

\[ \begin{cases} a_1x + b_1y = c_1 \\ a_2x + b_2y = c_2 \end{cases} \]

\[ \text{where } a_1, b_1, c_1, a_2, b_2, c_2 \text{ are constants} \]

\[ \text{Goal: Find values of } x \text{ and } y \text{ that satisfy both equations} \]

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System Definition and Graphical Representation

Consider the system of two linear equations with two variables x and y:

\[ \begin{cases} a_1x + b_1y = c_1 \\ a_2x + b_2y = c_2 \end{cases}, \quad \frac{a_1}{a_2} \neq \frac{b_1}{b_2} \]

\[ \text{Graphical representation shows two intersecting lines: } a_1x + b_1y = c_1 \text{ and } a_2x + b_2y = c_2 \]

\[ \text{Solution point } (x_1, y_1) \text{ represents intersection of both lines} \]

🎯 What does this mean?

A system of two linear equations consists of two linear equations with the same variables that must be satisfied simultaneously. The solution is the point (x, y) where both lines intersect on the coordinate plane. These systems model real-world situations involving multiple constraints, such as supply-demand equilibrium, mixture problems, and optimization scenarios with two variables and two conditions.

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Solution Formulas Using Determinants

Solution for x:

\[ x = \frac{\begin{vmatrix} c_1 & b_1 \\ c_2 & b_2 \end{vmatrix}}{\begin{vmatrix} a_1 & b_1 \\ a_2 & b_2 \end{vmatrix}} = \frac{c_1b_2 - c_2b_1}{a_1b_2 - a_2b_1} \]

Solution for y:

\[ y = \frac{\begin{vmatrix} a_1 & c_1 \\ a_2 & c_2 \end{vmatrix}}{\begin{vmatrix} a_1 & b_1 \\ a_2 & b_2 \end{vmatrix}} = \frac{a_1c_2 - a_2c_1}{a_1b_2 - a_2b_1} \]

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Terminology

\[ \text{System of Linear Equations: A set of two or more linear equations with the same variables.} \]

\[ \text{Determinant: A scalar value calculated from the coefficients, used to solve linear systems.} \]

\[ \text{Consistent System: When the system has at least one solution; here, the condition } \frac{a_1}{a_2} \neq \frac{b_1}{b_2} \text{ ensures a unique solution.} \]

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Applications

\[ \text{Used in algebra to find points of intersection of two lines.} \]

\[ \text{Fundamental in solving problems in physics involving simultaneous constraints.} \]

\[ \text{Applied in economics for optimization problems involving multiple variables.} \]

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Solution Methods Overview

Three primary approaches to solving linear systems:

\[ \text{1. Substitution Method: Solve for one variable, substitute} \]

\[ \text{2. Elimination Method: Add/subtract to eliminate variables} \]

\[ \text{3. Graphical Method: Find intersection point visually} \]

\[ \text{4. Matrix Method: Use inverse matrices or row operations} \]

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Substitution Method Steps

Systematic approach using substitution:

\[ \text{Step 1: Solve one equation for one variable} \]

\[ \text{Step 2: Substitute expression into other equation} \]

\[ \text{Step 3: Solve resulting single-variable equation} \]

\[ \text{Step 4: Back-substitute to find other variable} \]

\[ \text{Step 5: Check solution in both original equations} \]

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Elimination Method Steps

Systematic approach using elimination:

\[ \text{Step 1: Align equations with like terms vertically} \]

\[ \text{Step 2: Multiply equations to create opposite coefficients} \]

\[ \text{Step 3: Add equations to eliminate one variable} \]

\[ \text{Step 4: Solve for remaining variable} \]

\[ \text{Step 5: Substitute back to find other variable} \]

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Types of Solutions

Three possible outcomes for linear systems:

\[ \text{Unique Solution: Lines intersect at exactly one point} \]

\[ \text{No Solution: Parallel lines (inconsistent system)} \]

\[ \text{Infinite Solutions: Same line (dependent system)} \]

\[ \text{Determinant test: } D = a_1b_2 - a_2b_1 \]

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Cramer's Rule Method

Using determinants to solve systems:

\[ D = \begin{vmatrix} a_1 & b_1 \\ a_2 & b_2 \end{vmatrix} = a_1b_2 - a_2b_1 \]

\[ D_x = \begin{vmatrix} c_1 & b_1 \\ c_2 & b_2 \end{vmatrix} = c_1b_2 - c_2b_1 \]

\[ D_y = \begin{vmatrix} a_1 & c_1 \\ a_2 & c_2 \end{vmatrix} = a_1c_2 - a_2c_1 \]

\[ x = \frac{D_x}{D}, \quad y = \frac{D_y}{D} \quad \text{(if } D \neq 0\text{)} \]

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Matrix Representation

Linear algebra approach to systems:

\[ \begin{bmatrix} a_1 & b_1 \\ a_2 & b_2 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} c_1 \\ c_2 \end{bmatrix} \]

\[ A\vec{x} = \vec{b} \quad \text{(Matrix equation form)} \]

\[ \vec{x} = A^{-1}\vec{b} \quad \text{(If inverse exists)} \]

\[ \text{Augmented matrix: } \left[\begin{array}{cc|c} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \end{array}\right] \]

🎯 Mathematical Interpretation

Systems of two linear equations represent the intersection of two straight lines in the coordinate plane. The solution point (if it exists) satisfies both linear relationships simultaneously, modeling equilibrium conditions in real-world scenarios. These systems appear in economics (supply-demand equilibrium), physics (force balance), chemistry (mixture problems), and business (break-even analysis). The geometric interpretation helps visualize whether solutions exist and understand the relationships between variables.

\[ x, y \]

Variables - unknown values that must satisfy both equations simultaneously

\[ a_1, b_1, c_1 \]

First equation coefficients - constants defining the first linear relationship

\[ a_2, b_2, c_2 \]

Second equation coefficients - constants defining the second linear relationship

\[ (x, y) \]

Solution point - coordinate pair that satisfies both equations

\[ D = a_1b_2 - a_2b_1 \]

System determinant - value that determines solution type and existence

\[ \text{Substitution Method} \]

Solution technique - solve for one variable and substitute into other equation

\[ \text{Elimination Method} \]

Solution technique - add/subtract equations to eliminate variables

\[ \text{Consistent System} \]

Solution existence - system has at least one solution (unique or infinite)

\[ \text{Inconsistent System} \]

No solution case - equations represent parallel lines that never intersect

\[ \text{Dependent System} \]

Infinite solutions - equations represent the same line

\[ \text{Augmented Matrix} \]

Matrix representation - coefficient matrix with constants column attached

\[ \text{Cramer's Rule} \]

Determinant method - using ratios of determinants to find solutions

🎯 Essential Insight: Linear systems are like mathematical meeting points - they find where two straight-line relationships agree on the same values! 🤝

🚀 Real-World Applications

💰 Economics & Business

Supply-Demand & Break-Even Analysis

Economists use linear systems for market equilibrium analysis, break-even point calculations, profit optimization, and cost-revenue intersection modeling

🧪 Chemistry & Physics

Mixture Problems & Force Balance

Scientists apply linear systems for chemical mixture calculations, alloy composition problems, force equilibrium analysis, and conservation law applications

🚗 Engineering & Transportation

Network Flow & Resource Allocation

Engineers use linear systems for traffic flow optimization, resource distribution, project scheduling, and system capacity planning

📊 Finance & Investment

Portfolio Optimization & Budget Planning

Financial analysts apply linear systems for investment portfolio balancing, budget constraint analysis, risk-return optimization, and resource allocation

The Magic: Economics: Market equilibrium and break-even analysis, Science: Mixture problems and force balance calculations, Engineering: Network optimization and resource allocation, Finance: Portfolio balancing and budget constraint modeling

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Master the "Intersection Hunter" Method!

Before solving complex linear systems, develop this strategic approach:

Key Insight: Linear systems are like mathematical detectives hunting for the intersection point where two line "stories" agree. Each equation tells you a relationship between x and y, and the solution is where both stories are simultaneously true. Think of it as finding the exact spot where two roads cross!

💡 Why this matters:

🔋 Real-World Power:

Economics: Market equilibrium points and break-even analysis
Science: Chemical mixture calculations and force balance problems
Engineering: Network optimization and resource allocation
Finance: Investment portfolio balancing and budget constraints

🧠 Mathematical Insight:

Geometric meaning: Solution is intersection point of two lines
Three possibilities: One solution, no solution, or infinite solutions
Multiple methods: Substitution, elimination, graphical, matrix approaches
Verification crucial: Always check solutions in both original equations

🚀 Study Strategy:

1 Choose the Right Method 📐

Substitution: When one variable has coefficient 1 or -1
Elimination: When coefficients can be easily made opposites
Graphical: For visual understanding or rough estimates
Key insight: "Which method makes the algebra easiest?"

2 Apply Substitution Method 📋

Step 1: Solve easier equation for one variable
Step 2: Substitute expression into other equation
Step 3: Solve resulting single-variable equation
Step 4: Back-substitute to find other variable

3 Apply Elimination Method 🔗

Step 1: Multiply equations to create opposite coefficients
Step 2: Add equations to eliminate one variable
Step 3: Solve for remaining variable
Step 4: Substitute back to find eliminated variable

4 Verify and Interpret Solutions 🎯

Check: Substitute solution into both original equations
Interpret: What does solution mean in real-world context?
Special cases: Recognize parallel lines (no solution) or same line (infinite)
Apply: Use solutions to answer original problem questions

When you master the "intersection hunter" approach and understand linear systems as finding agreement points between relationships, you'll have powerful tools for solving equilibrium problems, mixture calculations, and optimization scenarios across economics, science, and engineering!

Memory Trick: "Two Lines, One Point" - SUBSTITUTION: Replace and solve, ELIMINATION: Add to remove, CHECK: Verify in both

🔑 Key Properties of Linear Systems

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Geometric Intersection

Solution represents intersection point of two lines in coordinate plane

Visual interpretation helps understand solution existence and uniqueness

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Three Solution Types

Unique solution (intersecting lines), no solution (parallel), infinite solutions (same line)

Determinant value predicts which case applies before solving

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Multiple Solution Methods

Substitution, elimination, graphical, and matrix approaches all valid

Method choice depends on equation structure and personal preference

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Real-World Modeling

Natural representation for equilibrium and constraint problems

Essential for optimization problems involving two variables and conditions

Universal Insight: Linear systems are mathematical intersection finders that locate the exact point where two linear relationships agree simultaneously!

General Form: Two equations with same variables solved simultaneously

Solution Methods: Substitution, elimination, graphical, and matrix approaches

Verification Rule: Always check solutions in both original equations

Applications: Market equilibrium, mixture problems, resource allocation, and optimization

System of Two Linear Equations – Solution Methods