5.4 Finding Linear Equations

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.

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.

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).

Solution Since the slope is , the equation has the form

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

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.

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.

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).

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

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

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

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.

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.

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.

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

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.

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

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.

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)

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.

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.