1. 5.4 Finding Linear Equations

2. Finding an Equation of a Line by Using the Slope, a Point, and the Slope-Intercept Form
To find an equation of a line by using the slope, a
point, and the slope-intercept form,
1. Substitute the given value of the slope m into the equation y = mx + b.
2. Substitute the coordinates of the given point into your equation from step 1, and solve for b.

3. Finding an Equation of a Line by Using the Slope, a Point, and the Slope-Intercept Form
3. Substitute the value of b from step 2 into your equation from step 1.
4. Check that the graph of your equation contains the given point.

4. Example: Using the Slope and a Point to Find a Linear Equation
Find an equation of the line that has slope m = and contains the point (−4, 1).

5. Solution
Since the slope is , the equation has the form

6. Solution So, the equation is
To find b, we substitute the coordinates of the point (−4, 1) into the equation
So, the equation is

7. to determine , to find the slope of the line
Finding an Equation of a Line by Using Two Points and the Slope-Intercept Form
To find an equation of the line that passes through two given points whose x -coordinates are different,
1. Use the formula , or use a graph
to determine , to find the slope of the line
containing the two points.

8. Finding an Equation of a Line by Using Two Points and the Slope-Intercept Form
2. Substitute the m value you found in step 1 into the equation y = mx + b.
3. Substitute the coordinates of one of the given points into the equation from step 2, and solve for b.
4. Substitute the b value you found in step 3 into your equation from step 2.
5. Check that the graph of your equation contains the two given points.

9. Example: Using Two Points to Find a Linear Equation
Find an equation of the line that passes through the points (−9, −2) and (−3, 7).

10. Solution
Find the slope of the line:
(−9, −2) and (−3, 7)
So, we have

11. Solution So, the equation is
To find b, we substitute the coordinates of the point (−3, 7) into the equation
So, the equation is

12. Example: Finding an Equation of a Vertical Line
Find an equation of the line that contains the points (5, 1) and (5, 3).

13. Solution
(5, 1) and (5, 3)
Since the x-coordinates of the given points are equal (both 5), the line that contains the points is vertical. An equation of the line is x = 5.

14. Example: Finding an Equation of a Line Parallel to a Given Line
Find an equation of the line l that contains the point (5, 3) and is parallel to the line y = 2x – 3.

15. Solution
For the line y = 2x – 3, the slope is 2. So, then slope of parallel line l is also 2. An equation of the line l is y = 2x + b.
To find b, substitute the coordinates of (5, 3) into the equation y = 2x + b:
The equation of l is y = 2x – 7.

16. We use a graphing calculator to verify our equation.
Solution
We use a graphing calculator to verify our equation.

17. Example: Finding an Equation of a Line Perpendicular to a Given Line
Find an equation of the line l that contains the point (4, 2) and is perpendicular to the line x + 3y = –6.

18. First, we write x + 3y = –6 in slope-intercept form:
Solution
First, we write x + 3y = –6 in slope-intercept form:
The slope is The slope of line l is

19. Solution An equation of line l is y = 3x – 10.
To find b, substitute the coordinates of the given point (4, 2) into y = 3x + b.
An equation of line l is y = 3x – 10.

20. Point-Slope Form of an Equation of a Line
If a nonvertical line has slope m and contains the point (x1, y1), then an equation of the line is
y − y1 = m(x − x1)

21. Example: Using the Point-Slope Form to Find an Equation of a Line
Use the point-slope form to find an equation of the line that has slope m = −3 and contains the point (−4, 2). Then write the equation in slope-intercept form.

22. Solution So, the equation is y = –3x − 10.
Substitute x1 = −4, y1 = 2, and m = −3 into the point-slope form y − y1 = m(x − x1):
So, the equation is y = –3x − 10.