**Equations of Straight Lines**

Depending upon the given information, equations of lines can take on several forms:

**Slope Intercept Form:**

y = mx + b

Use this form when you know, or can find, the slope, m, and the y-intercept, b.

**Point Slope Form:**

y – y_{1} = m(x – x_{1})

Use this form when you know, or can find, a point on the line (x_{1}, y_{1}), and the slope, m.

**Standard Form:**

Ax + By = C

The A and B values in this form cannot be zero. Use when asked to state the answer in Standard Form.

May also be Ax + By – C = 0.

**Horizontal Line Form:**

y = 7 (or any Real number)

Lines that are horizontal have a slope of zero. They have “run”, but no “rise”. The rise/run formula for slope always yields zero since rise = 0. Every point on this line has a y-value of 7. When writing the equation, we have

y = mx + b

y = 0x + 7

y = 7.

Note: The equation of the x-axis is y = 0.

**Vertical Line Form:**

x = -5 (or any Real number)

Lines that are vertical have no slope (it does not exist, undefined). They have “rise”, but no “run”. The rise/run formula for slope always has a zero denominator and is undefined. Every point on this line has an x-value of -5.

Note: The equation of the y-axis is x = 0.

Note:

Lines that are parallel have equal slopes.

Lines that are perpendicular have negative reciprocal slopes.

(A line with m = 4 will be perpendicular to a line with m = -ΒΌ)

**Examples for Equations of Lines**

Here are a few of the more common types of problems involving the equations of lines.

**Example 1:** Find the slope and y-intercept of the equation 2y = 8x – 11.

Solution: First, solve for “y =”.

y = 4x – 5.5

y = mx + b

The slope, m, is 4.

The y-intercept, b, is -5.5.

**Example 2:** Find the equation of a line whose slope is -2 and who crosses the y-axis at (0,-3).

**Solution:** The m = -2 and b = -3.

y = mx + b

y = -2x + (-3)

y = -2x – 3

**Example 3:** Find the equation of a line whose slope is 4 and passes through the point (-3,5).

**Solution:** The m = 4 and (x_{1}, y_{1}) = (-3,5).

Point-Slope Form

y – y_{1} = m(x – x_{1})

y – 5 = 4(x – (-3))

y – 5 = 4(x + 3)

y – 5 = 4x + 12

y = 4x + 17

**Example 4:** Find the equation of a line passing through the points (-2,6) and (-4,-2).

**Solution:** Find slope first.

m = (6 – (-2))/(-2 – (-4)) = 8/2 = 4

Use either point as (x_{1}, y_{1}): (-2,6).

Point-Slope Form

y – y_{1} = m(x – x_{1})

y – 6 = 4(x – (-2))

y – 6 = 4(x + 2)

y – 6 = 4x + 8

y = 4x + 14

**Example 5:** Find the equation of a line that is parallel to the line y = -2x + 8 and passes through the point (3,6).

**Solution:** Parallel means equal slopes.

So, m = -2 and (x_{1}, y_{1}) = (3,6).

Point-Slope Form

y – y_{1} = m(x – x_{1})

y – 6 = -2(x – 3)

y – 6 = -2x + 6)

y = -2x + 12

**Example 6:** Find the equation of a line that is perpendicular to the line y = x + 7 and has the same y-intercept as 3y = 2x – 9. State the answer in Standard Form.

**Solution:** Perpendicular means negative reciprocal slopes.

So, m = -1 and b = -3.

y = mx + b

y = -1x + (-3)

y = -x – 3

x + y = -3 (Standard Form)