Linear Equations: The Foundation of Algebraic Problem Solving

What are Linear Equations?

A linear equation is an algebraic equation where each term is either a constant or the product of a constant and a single variable raised to the first power. Linear equations graph as straight lines, hence the name “linear.”

Linear Equation Characteristics
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Standard Forms:
One variable: ax + b = c
Two variables: ax + by = c
Slope-intercept: y = mx + b

Key Properties:
✓ Variables have degree 1 (first power only)
✓ No variables in denominators
✓ No variables under radicals
✓ No variables as exponents
✓ Graph as straight lines

Examples:
Linear: 3x + 7 = 19, 2x - 5y = 10, y = 4x - 3
Not Linear: x² + 3 = 7, 1/x = 5, √x = 4

Solving Linear Equations in One Variable

Basic Solution Process

The goal is to isolate the variable on one side of the equation using inverse operations.

Solution Strategy
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1. Simplify both sides (distribute, combine like terms)
2. Move variable terms to one side
3. Move constant terms to the other side
4. Divide by the coefficient of the variable
5. Check your solution

Example: 3(x + 2) - 5 = 2x + 7

Step 1: Distribute and simplify
3x + 6 - 5 = 2x + 7
3x + 1 = 2x + 7

Step 2: Move variable terms
3x - 2x = 7 - 1
x = 6

Step 3: Check
3(6 + 2) - 5 = 3(8) - 5 = 24 - 5 = 19
2(6) + 7 = 12 + 7 = 19 ✓

Types of Linear Equation Solutions

Solution Types
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One Solution (Most Common):
2x + 3 = 7
x = 2
The equation has exactly one value that makes it true

No Solution (Contradiction):
2x + 3 = 2x + 5
3 = 5 (impossible)
The equation is never true

Infinite Solutions (Identity):
2x + 3 = 2x + 3
0 = 0 (always true)
Every value of x makes the equation true

Applications of Linear Equations

Word Problems and Real-World Applications

Linear equations model many real-world situations involving constant rates of change.

Common Application Types
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Age Problems:
"John is 5 years older than Mary. In 3 years, their combined age will be 35."
Let x = Mary's current age
Then x + 5 = John's current age
Equation: (x + 3) + (x + 5 + 3) = 35

Distance-Rate-Time Problems:
"A car travels 240 miles in 4 hours. What is its speed?"
d = rt → 240 = r(4) → r = 60 mph

Mixture Problems:
"How many pounds of $8/lb coffee should be mixed with 5 pounds of $12/lb coffee to get a mixture worth $10/lb?"
Let x = pounds of $8 coffee
8x + 12(5) = 10(x + 5)

Money Problems:
"Sarah has $3.50 in quarters and dimes. She has 5 more quarters than dimes. How many of each coin does she have?"
Let d = number of dimes
Then d + 5 = number of quarters
0.10d + 0.25(d + 5) = 3.50

Percent Problems

Percent Applications
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Basic Percent Equation: Part = Percent × Whole
Or: P = r × W

Percent Increase/Decrease:
New Amount = Original ± (Percent × Original)
A = P ± rP = P(1 ± r)

Examples:
1. "What is 15% of 80?"
   P = 0.15 × 80 = 12

2. "25 is what percent of 200?"
   25 = r × 200 → r = 0.125 = 12.5%

3. "30% of what number is 18?"
   18 = 0.30 × W → W = 60

4. "A $200 item is marked up 25%. What's the new price?"
   New price = 200(1 + 0.25) = 200(1.25) = $250

Linear Equations in Two Variables

Graphing Linear Equations

Linear equations in two variables represent relationships between two quantities and graph as straight lines.

Graphing Methods
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Method 1: Table of Values
For y = 2x + 1:
x | y = 2x + 1 | (x,y)
-2| 2(-2)+1=-3| (-2,-3)
-1| 2(-1)+1=-1| (-1,-1)
 0| 2(0)+1=1  | (0,1)
 1| 2(1)+1=3  | (1,3)
 2| 2(2)+1=5  | (2,5)

Method 2: Intercepts
x-intercept: Set y = 0, solve for x
y-intercept: Set x = 0, solve for y

For 2x + 3y = 12:
x-intercept: 2x + 3(0) = 12 → x = 6 → (6,0)
y-intercept: 2(0) + 3y = 12 → y = 4 → (0,4)

Method 3: Slope-Intercept Form
y = mx + b
m = slope, b = y-intercept
Start at (0,b), use slope to find next points

Slope and Rate of Change

Understanding Slope
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Definition: Slope = rise/run = (y₂ - y₁)/(x₂ - x₁)

Slope Interpretation:
m > 0: Line rises from left to right
m < 0: Line falls from left to right
m = 0: Horizontal line
m undefined: Vertical line

Rate of Change:
Slope represents the rate at which y changes with respect to x

Examples:
1. Points (1,3) and (4,9):
   m = (9-3)/(4-1) = 6/3 = 2

2. y = -3x + 7:
   Slope = -3 (for every 1 unit right, go 3 units down)

3. Real-world: "Water drains from a tank at 5 gallons per minute"
   Slope = -5 (negative because water is decreasing)

Systems of Linear Equations

Solving Systems by Substitution

Substitution Method
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Steps:
1. Solve one equation for one variable
2. Substitute into the other equation
3. Solve for the remaining variable
4. Back-substitute to find the first variable
5. Check the solution

Example:
{2x + y = 7
{x - y = 2

Step 1: From equation 2: x = y + 2
Step 2: Substitute into equation 1:
        2(y + 2) + y = 7
        2y + 4 + y = 7
        3y = 3
        y = 1
Step 3: Back-substitute: x = 1 + 2 = 3
Step 4: Check: 2(3) + 1 = 7 ✓, 3 - 1 = 2 ✓
Solution: (3, 1)

Solving Systems by Elimination

Elimination Method
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Steps:
1. Align equations vertically
2. Multiply equations to create opposite coefficients
3. Add equations to eliminate one variable
4. Solve for remaining variable
5. Back-substitute
6. Check solution

Example:
{3x + 2y = 16
{5x - 2y = 8

Step 1: Coefficients of y are already opposites
Step 2: Add equations:
        3x + 2y = 16
        5x - 2y = 8
        ___________
        8x = 24
        x = 3

Step 3: Substitute x = 3 into first equation:
        3(3) + 2y = 16
        9 + 2y = 16
        2y = 7
        y = 3.5

Solution: (3, 3.5)

Types of Systems

System Solution Types
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One Solution (Consistent and Independent):
Lines intersect at exactly one point
Different slopes: m₁ ≠ m₂

No Solution (Inconsistent):
Lines are parallel (never intersect)
Same slope, different y-intercepts: m₁ = m₂, b₁ ≠ b₂

Infinite Solutions (Consistent and Dependent):
Lines are identical (same line)
Same slope and y-intercept: m₁ = m₂, b₁ = b₂

Graphical Interpretation:
One solution: Lines cross
No solution: Parallel lines
Infinite solutions: Same line

Linear Inequalities

Solving Linear Inequalities

Linear inequalities are solved similarly to equations, with one important difference: when multiplying or dividing by a negative number, the inequality sign flips.

Inequality Solution Rules
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Same as equations:
✓ Add/subtract same number to both sides
✓ Multiply/divide by positive number

Different from equations:
⚠ When multiplying/dividing by negative number,
  flip the inequality sign

Examples:
1. 3x + 5 > 14
   3x > 9
   x > 3

2. -2x + 7 ≤ 15
   -2x ≤ 8
   x ≥ -4  (sign flipped!)

3. -3(x - 2) < 9
   -3x + 6 < 9
   -3x < 3
   x > -1  (sign flipped!)

Graphing Linear Inequalities

Graphing on Number Line
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x > 3: Open circle at 3, arrow right
x ≥ 3: Closed circle at 3, arrow right
x < 3: Open circle at 3, arrow left
x ≤ 3: Closed circle at 3, arrow left

Compound Inequalities:
-2 < x ≤ 5: Open at -2, closed at 5, between them
x < -1 or x > 3: Two separate regions

Graphing in Coordinate Plane:
y > 2x + 1: Dashed line, shade above
y ≤ -x + 3: Solid line, shade below

Test Point Method:
1. Graph the boundary line
2. Choose test point not on line
3. Substitute into inequality
4. If true, shade that side; if false, shade other side

Problem-Solving Strategies

Setting Up Linear Equations

Problem-Solving Framework
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1. READ and UNDERSTAND
   - What information is given?
   - What are we asked to find?
   - What are the relationships?

2. DEFINE VARIABLES
   - Choose meaningful variable names
   - State what each variable represents
   - Include units if applicable

3. WRITE EQUATIONS
   - Translate words to mathematical expressions
   - Use given relationships
   - Check that units are consistent

4. SOLVE
   - Use appropriate algebraic techniques
   - Show all steps clearly
   - Check solution in original equation

5. INTERPRET and VERIFY
   - Does the answer make sense?
   - Check against original problem
   - Include appropriate units

Common Problem Types

Consecutive Integer Problems
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Consecutive integers: n, n+1, n+2, ...
Consecutive even: n, n+2, n+4, ... (n even)
Consecutive odd: n, n+2, n+4, ... (n odd)

Example: "Find three consecutive integers whose sum is 48."
Let n = first integer
Then n+1 = second, n+2 = third
Equation: n + (n+1) + (n+2) = 48
Solution: 3n + 3 = 48 → n = 15
Answer: 15, 16, 17

Geometry Problems
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Perimeter: P = sum of all sides
Area formulas: Rectangle A = lw, Triangle A = ½bh

Example: "A rectangle's length is 3 more than twice its width. If the perimeter is 36, find the dimensions."
Let w = width
Then 2w + 3 = length
Equation: 2w + 2(2w + 3) = 36
Solution: 2w + 4w + 6 = 36 → w = 5
Answer: width = 5, length = 13

Linear Functions and Modeling

Function Notation and Linear Functions

Linear Function Properties
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Function Notation: f(x) = mx + b
- f(x) represents the output (y-value)
- x represents the input
- m is the slope (rate of change)
- b is the y-intercept (initial value)

Domain and Range:
- Domain: all real numbers (unless restricted)
- Range: all real numbers (unless restricted)

Key Features:
✓ Constant rate of change (slope)
✓ Straight line graph
✓ One-to-one correspondence (passes vertical line test)

Examples:
f(x) = 2x + 3: slope = 2, y-intercept = 3
g(x) = -½x + 1: slope = -½, y-intercept = 1
h(x) = 5: slope = 0, horizontal line at y = 5

Linear Modeling

Real-World Linear Models
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Cost Functions:
C(x) = mx + b
m = variable cost per unit
b = fixed cost

Example: "A company has fixed costs of $500 and variable costs of $25 per item."
C(x) = 25x + 500

Revenue Functions:
R(x) = px (where p = price per unit)

Profit Functions:
P(x) = R(x) - C(x)

Break-even Point:
Set R(x) = C(x) and solve for x

Temperature Conversion:
F = (9/5)C + 32
C = (5/9)(F - 32)

Population Growth (linear):
P(t) = P₀ + rt
P₀ = initial population
r = rate of change per time unit

Summary and Key Concepts

Linear equations form the foundation of algebraic problem-solving and provide essential tools for modeling real-world relationships. Mastering linear equations prepares you for more advanced algebraic concepts and applications.

Chapter Summary
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Essential Skills Mastered:
✓ Solving linear equations in one variable
✓ Graphing linear equations and inequalities
✓ Solving systems of linear equations
✓ Applying linear equations to word problems
✓ Understanding slope and rate of change
✓ Working with linear functions and modeling

Key Formulas:
• Standard form: ax + by = c
• Slope-intercept form: y = mx + b
• Point-slope form: y - y₁ = m(x - x₁)
• Slope formula: m = (y₂ - y₁)/(x₂ - x₁)
• Distance formula: d = √[(x₂-x₁)² + (y₂-y₁)²]

Problem-Solving Tools:
• Substitution and elimination methods
• Graphical interpretation
• Function notation and modeling
• Real-world applications

Next Steps:
These linear equation skills provide the foundation for:
- Quadratic equations and functions
- Polynomial operations
- Exponential and logarithmic functions
- Advanced algebraic concepts

Linear equations represent the gateway to algebraic thinking, providing both computational tools and conceptual frameworks that extend throughout mathematics. The skills developed in this chapter - systematic problem-solving, graphical interpretation, and mathematical modeling - will serve as essential building blocks for all future algebraic learning.