Mastering Linear Equations: Forms, Slope, and Intercepts

Posted on Mar 7, 2025 in Mathematics

Forms of Linear Equations

1. Slope-Intercept Form

This form is represented as:

y = mx + b

• m = slope (rise/run)
• b = where the line crosses the y-axis (y-intercept)

How to Use:

1. Start at b (on the y-axis).
2. Use the slope (rise/run) to plot more points.

Example:
If m = 2 and b = -3, the equation is:
y = 2x – 3

2. Point-Slope Form

This form uses:

y – y₁ = m(x – x₁)

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

Steps to Convert to Slope-Intercept Form:

1. Start with the formula: y – y₁ = m(x – x₁)
2. Distribute (multiply) the slope.
3. Solve for y to make it look like y = mx + b.

Example:
Slope m = 3, Point (2, 5):
Start with: y – 5 = 3(x – 2)
Distribute: y – 5 = 3x – 6
Solve for y: y = 3x – 1

3. General Form

This form is represented as:

Ax + By + C = 0

• A, B, and C are numbers.
• A must be positive (no fractions).

How to Convert Slope-Intercept to General Form:

1. Start with y = mx + b.
2. Move everything to one side so it equals 0.

Example:
Convert y = -2x + 4 to general form:
Move -2x to the left: 2x + y – 4 = 0

Finding the Slope

The slope (m) shows how steep a line is.

Formula:
m = (y₂ – y₁) / (x₂ – x₁)

Steps:

1. Pick two points: (x₁, y₁) and (x₂, y₂).
2. Subtract the y-values (rise).
3. Subtract the x-values (run).
4. Divide.

Example:
Find the slope between (1, 2) and (3, 6):
m = (6 – 2) / (3 – 1) = 4 / 2 = 2

X- and Y-Intercepts

Y-Intercept

The y-intercept is where the line crosses the y-axis.

• Set x = 0 and solve for y.

Example: For y = 2x – 4:
Set x = 0: y = 2(0) – 4 = -4
Y-intercept: (0, -4)

X-Intercept

The x-intercept is where the line crosses the x-axis.

• Set y = 0 and solve for x.

Example: For y = 2x – 4:
Set y = 0: 0 = 2x – 4 → x = 2
X-intercept: (2, 0)

Solving Systems of Linear Equations

A system is when you have 2 lines. The solution is where they meet.

Ways to Solve:

1. Graphing: Graph both lines and look for where they meet.
2. Substitution: Solve for one variable and plug it into the other equation.
3. Elimination: Add or subtract to remove one variable.

Example (Substitution):
Solve: y = 2x + 1 and y = -x + 4

1. Substitute 2x + 1 into the second equation:
   2x + 1 = -x + 4
2. Solve for x:
   3x = 3 → x = 1
3. Solve for y:
   y = 2(1) + 1 = 3
   Solution: (1, 3)

Key Formulas to Remember

1. Slope Formula:
   m = (y₂ – y₁) / (x₂ – x₁)

2. Slope-Intercept Form:
   y = mx + b

3. Point-Slope Form:
   y – y₁ = m(x – x₁)

4. General Form:
   Ax + By + C = 0

5. X-Intercept: Set y = 0.

6. Y-Intercept: Set x = 0.

1. General Form to Slope-Intercept Form

General Form:

Ax + By + C = 0

Slope-Intercept Form:

y = mx + b

Steps to Change:

1. Move Ax and C to the other side.
2. Divide by B to solve for y.

Example:
Start with 2x + 3y – 6 = 0.

1. Move 2x and -6 to the right:
   3y = -2x + 6

2. Divide everything by 3:
   y = (-2/3)x + 2

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

2. Slope-Intercept Form to General Form

Slope-Intercept Form:

y = mx + b

General Form:

Ax + By + C = 0

Steps to Change:

1. Move mx to the left side.
2. Make sure A (number with x) is positive.

Example:
Start with y = -2x + 4.

1. Move -2x to the left:
   2x + y – 4 = 0

Answer: 2x + y – 4 = 0

3. Point-Slope Form to General Form

Point-Slope Form:

y – y₁ = m(x – x₁)

General Form:

Ax + By + C = 0

Steps to Change:

1. Distribute (multiply) the slope.
2. Move everything to one side so it equals 0.

Example:
Start with y – 3 = 2(x – 1).

1. Distribute 2 to (x – 1):
   y – 3 = 2x – 2

2. Move 2x and -2 to the left:
   -2x + y – 3 + 2 = 0

3. Combine like terms:
   -2x + y – 1 = 0

4. Make x positive (multiply by -1):
   2x – y + 1 = 0

Answer: 2x – y + 1 = 0

4. Point-Slope Form to Slope-Intercept Form

Point-Slope Form:

y – y₁ = m(x – x₁)

Slope-Intercept Form:

y = mx + b

Steps to Change:

1. Distribute (multiply) the slope.
2. Solve for y.

Example:
Start with y – 3 = 2(x – 1).

1. Distribute 2 to (x – 1):
   y – 3 = 2x – 2

2. Add 3 to both sides:
   y = 2x – 2 + 3

3. Combine like terms:
   y = 2x + 1

Answer: y = 2x + 1