There are three different scenarios that may require you to find a linear equation. Learn them and you’ll definitely improve your grade on this topic.

Usually, the slope-intercept form is used: y = mx +b

(Remember “m” represents the slope, and “b” represents the y-intercept)

Just remember, linear equations require at least two things:

**a slope and a point. If not given, these two items will help us find the slope and y-intercept.**

**I. Find the Equation of a Line**Given a Slope and a Y-intercept

Example: Given m = -2 and y-intercept = 4

Think: In order to write a linear equation, we need a slope and a point.

The slope is 2

The y-intercept 4 is the point (0, 4).

1) Use the slope-intercept form of a line: y = mx + b

2) Substitute values into equation

3) The answer is y = -2X + 4.

**II. Find the Equation of a Line Given**a Slope and a Point

Example: Find the equation of a line with a slope of -3 and passes through (4, 6)

Or the problem may be presented as: “Given m = -3 and (4, 6) find the line”

Think: I want to write a linear equation in the form of y = mx + b

The slope is -3

The point is (4, 6). Note: this is a point the line passes through, but it is NOT the y-intercept. So,

1) Use the slope-intercept form of a line: y = mx + b

2) Substitute values into equation (remember “m”

represents the slope, and “b” represents the y-intercept) The y-intercept was not given in this problem. However, we were given the point, (4, 6). Thus X=4 and Y=6

Substitute the values as follows to find the y-intercept, “b”.

y = mx + b

6 = -3(4) + b

6 = -12 + b

6 + 12 = -12 + 12 + b

18 = b

3) Thus, the answer is y = -3X + 18

**III. Find the Equation of a Line**Given the Line Passes Through Two Points.

**Example: Find the line that passes through (3, 6) and (7, 26)**

Think: I want to write a linear equation in the form of y = mx + b

The slope – not given

Y-intercept – not given

So, I’ll do the following.

1) Use the two points to find the slope.

2) Use the slope and either one of the points to plug into the equation, y = mx + b, to find the value of b, the y-intercept.

Step 1: slope = (26 – 6) / (7 – 3) = 20/4 = 5

Step 2: m = 5 and let’s choose to use the point (3, 6)

Y = mx + b

6 = 5(3) + b

6 = 15 + b

6 – 15 = 15- 15 + b

-9 = b

Thus, the answer is y= 5X – 9

Note: If you need help finding the slope, please refer to link below