Linear Equations

Linear equations are the foundation of algebra! They represent straight-line relationships and appear everywhere in the real world. Understanding how to write, graph, and use linear equations opens doors to advanced mathematics and practical problem-solving.

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What Is a Linear Equation?

A linear equation is an equation whose graph is a straight line.

General form: Ax + By = C (where A, B, C are constants)

Most common form: y = mx + b (slope-intercept form)

Characteristics:

• Variables have exponent of 1 (no x², x³, etc.)
• Graph is always a straight line
• Can have one or two variables

Examples of linear equations:

• y = 2x + 3
• 3x + 4y = 12
• y = -5x
• x = 7
• y = 4

NOT linear equations:

• y = x² (parabola)
• y = 1/x (hyperbola)
• xy = 6 (hyperbola)

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Slope-Intercept Form

y = mx + b

Where:

• m = slope (steepness and direction)
• b = y-intercept (where line crosses y-axis)

This is the most useful form for graphing and understanding lines!

Example 1: y = 3x + 2

• Slope (m) = 3
• Y-intercept (b) = 2
• Line rises 3 units for every 1 unit right
• Crosses y-axis at (0, 2)

Example 2: y = -2x + 5

• Slope (m) = -2
• Y-intercept (b) = 5
• Line falls 2 units for every 1 unit right
• Crosses y-axis at (0, 5)

Example 3: y = (1/2)x - 3

• Slope (m) = 1/2
• Y-intercept (b) = -3
• Line rises 1 unit for every 2 units right
• Crosses y-axis at (0, -3)

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Converting to Slope-Intercept Form

Goal: Solve for y to get y = mx + b

Example 1: Convert 2x + y = 8 to slope-intercept form

Solution: 2x + y = 8 Subtract 2x from both sides: y = -2x + 8

Answer: y = -2x + 8

• Slope = -2
• Y-intercept = 8

Example 2: Convert 3x - 2y = 6 to slope-intercept form

Solution: 3x - 2y = 6 Subtract 3x: -2y = -3x + 6 Divide by -2: y = (3/2)x - 3

Answer: y = (3/2)x - 3

• Slope = 3/2
• Y-intercept = -3

Example 3: Convert 4x + 2y = 10 to slope-intercept form

Solution: 4x + 2y = 10 -2y = -4x + 10 y = 2x - 5

Answer: y = 2x - 5

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Graphing Linear Equations

Method 1: Using Slope and Y-Intercept

Steps:

1. Identify b (y-intercept) and plot point (0, b)
2. Use slope m = rise/run to find another point
3. Draw a line through the points

Example: Graph y = 2x + 1

Step 1: Y-intercept = 1, plot (0, 1)

Step 2: Slope = 2 = 2/1 (rise 2, run 1) From (0, 1): go up 2, right 1 → (1, 3)

Step 3: Draw line through (0, 1) and (1, 3)

Method 2: Using Two Points (Table of Values)

Steps:

1. Choose x-values (usually include 0)
2. Calculate corresponding y-values
3. Plot points
4. Draw line

Example: Graph y = -x + 4

| x | y = -x + 4 | Point | |---|-----------|-------| | 0 | -0 + 4 = 4 | (0, 4) | | 2 | -2 + 4 = 2 | (2, 2) | | 4 | -4 + 4 = 0 | (4, 0) |

Plot points and draw line through them.

Method 3: Using Intercepts

Steps:

1. Find x-intercept (set y = 0)
2. Find y-intercept (set x = 0)
3. Plot both intercepts
4. Draw line

Example: Graph 2x + 3y = 6

X-intercept: Set y = 0 2x + 3(0) = 6 2x = 6 x = 3 → Point (3, 0)

Y-intercept: Set x = 0 2(0) + 3y = 6 3y = 6 y = 2 → Point (0, 2)

Plot (3, 0) and (0, 2), draw line.

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Writing Linear Equations

Given slope and y-intercept:

Simply plug into y = mx + b!

Example: Write equation with slope 4 and y-intercept -3

Answer: y = 4x - 3

Given slope and a point:

Use point-slope form: y - y₁ = m(x - x₁)

Example: Write equation with slope 2 passing through (3, 5)

Solution: y - 5 = 2(x - 3) y - 5 = 2x - 6 y = 2x - 1

Answer: y = 2x - 1

Given two points:

Step 1: Find slope using m = (y₂ - y₁)/(x₂ - x₁) Step 2: Use point-slope form with either point

Example: Write equation through (1, 3) and (4, 9)

Solution:

Step 1: Find slope m = (9 - 3)/(4 - 1) = 6/3 = 2

Step 2: Use point-slope form with (1, 3) y - 3 = 2(x - 1) y - 3 = 2x - 2 y = 2x + 1

Answer: y = 2x + 1

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Special Linear Equations

Horizontal Lines: y = k (constant)

• Slope = 0
• Parallel to x-axis
• Example: y = 5

Vertical Lines: x = k (constant)

• Undefined slope
• Parallel to y-axis
• NOT a function!
• Example: x = -2

Lines through Origin: y = mx

• Y-intercept = 0
• Passes through (0, 0)
• Example: y = 3x

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Parallel and Perpendicular Lines

Parallel Lines:

• Have the SAME slope
• Never intersect
• Different y-intercepts

Example: y = 2x + 3 and y = 2x - 5 are parallel (both have slope = 2)

Perpendicular Lines:

• Slopes are negative reciprocals
• Intersect at 90° angle
• If one slope is m, other is -1/m

Example: y = 3x + 1 and y = (-1/3)x + 4 are perpendicular

• Slopes: 3 and -1/3
• Product: 3 × (-1/3) = -1 ✓

Finding Parallel Line:

Example: Write equation parallel to y = 2x + 5 passing through (1, 4)

Solution: Same slope: m = 2 Use point-slope: y - 4 = 2(x - 1) y = 2x + 2

Answer: y = 2x + 2

Finding Perpendicular Line:

Example: Write equation perpendicular to y = 4x - 1 passing through (2, 3)

Solution: Original slope: 4 Perpendicular slope: -1/4 Use point-slope: y - 3 = (-1/4)(x - 2) y - 3 = (-1/4)x + 1/2 y = (-1/4)x + 7/2

Answer: y = (-1/4)x + 3.5

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Point-Slope Form

y - y₁ = m(x - x₁)

Where:

• m = slope
• (x₁, y₁) = a point on the line

Useful when you know:

• The slope
• One point on the line

Example: Write equation with slope -3 through point (2, 5)

Solution: y - 5 = -3(x - 2)

Can leave in this form or convert to slope-intercept: y - 5 = -3x + 6 y = -3x + 11

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Standard Form

Ax + By = C

Where A, B, C are integers (A should be positive)

Converting from slope-intercept to standard form:

Example: Convert y = 2x + 3 to standard form

Solution: y = 2x + 3 Subtract 2x from both sides: -2x + y = 3 Multiply by -1 to make A positive: 2x - y = -3

Answer: 2x - y = -3

Note: Standard form is useful for finding intercepts quickly!

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Real-World Applications

Cost Equations:

Problem: A taxi charges $3 plus$2 per mile. Write an equation for total cost.

Solution: Let x = miles traveled, y = total cost Fixed charge: $3 (y-intercept) Per mile:$2 (slope)

Equation: y = 2x + 3

Temperature Conversion:

Celsius to Fahrenheit: F = (9/5)C + 32

• Slope: 9/5
• Y-intercept: 32

Savings:

Problem: You have $50 and save$10 per week. Write equation for savings.

Solution: Starting amount: $50 (y-intercept) Weekly savings:$10 (slope)

Equation: y = 10x + 50 (where x = weeks, y = total)

Phone Plans:

Plan A: $20/month +$0.10 per text Equation: y = 0.10x + 20

Plan B: $30/month, unlimited texts Equation: y = 30 (horizontal line!)

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Solving Real-World Problems

Example: A plant is 6 cm tall and grows 2 cm per day.

a) Write equation for height

h = 2d + 6 (where d = days, h = height)

b) How tall after 10 days?

h = 2(10) + 6 = 26 cm

c) When will it be 20 cm tall?

20 = 2d + 6 14 = 2d d = 7 days

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Identifying Linear vs. Non-Linear

Linear (constant rate of change):

• y = 3x + 2 ✓
• 2x + y = 5 ✓
• y = 7 ✓
• Table with constant slope ✓

NOT Linear:

• y = x² (rate of change varies)
• y = 2ˣ (exponential)
• xy = 10 (variables multiply)
• y = 1/x (rational function)

Check a table:

| x | y | Change | |---|---|--------| | 1 | 5 | - | | 2 | 7 | +2 | | 3 | 9 | +2 | | 4 | 11 | +2 |

Constant change = LINEAR! ✓

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Common Mistakes to Avoid

❌ Mistake 1: Confusing slope and y-intercept

• In y = 3x + 5, slope is 3 (not 5!)

❌ Mistake 2: Sign errors when converting

• Wrong: 2x + y = 8 → y = 2x - 8
• Right: 2x + y = 8 → y = -2x + 8

❌ Mistake 3: Thinking vertical lines are y = k

• Wrong: Vertical line through x = 3 is y = 3
• Right: x = 3

❌ Mistake 4: Forgetting negative reciprocal for perpendicular

• Wrong: Perpendicular to m = 2 is m = -2
• Right: Perpendicular to m = 2 is m = -1/2

❌ Mistake 5: Not simplifying slope

• Not simplified: m = 6/8
• Simplified: m = 3/4

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Problem-Solving Strategy

For graphing:

1. Convert to y = mx + b if needed
2. Plot y-intercept
3. Use slope to find more points
4. Draw line

For writing equations:

1. Identify what you know (slope? points? intercept?)
2. Choose appropriate form
3. Substitute values
4. Simplify if needed

For applications:

1. Identify variables
2. Find slope (rate of change)
3. Find y-intercept (starting value)
4. Write equation
5. Use it to answer questions

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Quick Reference

Forms of Linear Equations:

Slope-Intercept: y = mx + b

• m = slope, b = y-intercept
• Best for graphing

Point-Slope: y - y₁ = m(x - x₁)

• Best when you know slope and a point

Standard Form: Ax + By = C

• Useful for finding intercepts

Slope Formula: m = (y₂ - y₁)/(x₂ - x₁)

Special Lines:

• Horizontal: y = k (slope = 0)
• Vertical: x = k (undefined slope)

Parallel Lines: Same slope

Perpendicular Lines: Slopes multiply to -1

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Practice Tips

Tip 1: Always identify m and b first

• Slope tells you steepness and direction
• Y-intercept tells you starting point

Tip 2: Check your graph

• Does it look steep enough?
• Does it cross y-axis at right spot?
• Do your points fit the line?

Tip 3: Use (0, 0) to check

• Plug x = 0, y = 0 into equation
• If it works, line passes through origin!

Tip 4: Verify with a third point

• After graphing, pick an x-value
• Calculate y using equation
• Check if it's on your line

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Summary

Linear equations represent straight-line relationships:

Key forms:

• Slope-intercept: y = mx + b (most common)
• Point-slope: y - y₁ = m(x - x₁)
• Standard form: Ax + By = C

Important concepts:

• Slope (m) shows rate of change
• Y-intercept (b) shows starting value
• Parallel lines have same slope
• Perpendicular lines have negative reciprocal slopes

Graphing methods:

• Use slope and y-intercept
• Plot points from table
• Use x and y intercepts

Real-world applications:

• Cost calculations
• Growth/decay over time
• Conversions
• Comparisons

Linear equations are fundamental to algebra and appear constantly in mathematics, science, business, and everyday life!