3 5 Equations of Lines SlopeIntercept Form of

§ 3. 5 Equations of Lines

Slope-Intercept Form of a Line Slope-Intercept Form When a linear equation in two variables is written in the slope-intercept form, y = mx + b m is the slope and (0, b) is the y-intercept of the line. y = 3 x – 4 y The slope The -intercept is is 3. (0, -4). Martin-Gay, Beginning and Intermediate Algebra, 4 ed 2

Slope-Intercept Form Example: Find the slope and y-intercept of the line – 3 x + y = – 5. First, we need to solve the linear equation for y. By adding 3 x to both sides, y = 3 x – 5. Once we have the equation in the form of y = mx + b, we can read the slope and y-intercept. slope is 3 y-intercept is (0, – 5) Martin-Gay, Beginning and Intermediate Algebra, 4 ed 3

Slope-Intercept Form Example: Find the slope and y-intercept of the line 2 x – 6 y = 12. First, we need to solve the linear equation for y. – 6 y = – 2 x + 12 Subtract 2 x from both sides. y= x– 2 Divide both sides by – 6. Since the equation is now in the form of y = mx + b, slope is y-intercept is (0, – 2) Martin-Gay, Beginning and Intermediate Algebra, 4 ed 4

The Point-Slope Form of the Equation of a Line The point-slope form of the equation of a line is slope (x 1, y 1) point on the line where m is the slope of the line and (x 1, y 1) is a point on the line. Martin-Gay, Beginning and Intermediate Algebra, 4 ed 5

The Point-Slope Form Example: Find an equation of the line whose slope is 5 and contains the point (4, 3). Write the equation in slope-intercept form. m = 5, x 1 = 4, y 1 = 3 y – y 1 = m(x – x 1) y – (– 3) = 5(x – 4) y + 3 = 5 x – 20 y = 5 x – 23 Substitute the values for m, x 1, and y 1. Simplify and distribute. Subtract 3 from both sides. Martin-Gay, Beginning and Intermediate Algebra, 4 ed 6

Finding the Equation Given Two Points Example: Find an equation of the line that passes through ( 2, 1) and (7, 4). Write the equation in slope-intercept form. Find the slope of the line. Use the point-slope form to find the equation. Continued. Martin-Gay, Beginning and Intermediate Algebra, 4 ed 7

Finding the Equation Given Two Points Example continued: Distribute. Add 1 to each side. Simplify. Martin-Gay, Beginning and Intermediate Algebra, 4 ed 8

Using the Point-Slope Form Example: In 1990, Window World , Inc. had 50 employees. In 2005, the company had 85 employees. Let x represent the number of years after 1990 and let y represent the number of employees. a. ) Assume that the relationship between years and number of employees is linear, and write an equation describing this relationship. b. ) Use the equation to predict the number of employees in 2000. Continued. Martin-Gay, Beginning and Intermediate Algebra, 4 ed 9

Using the Point-Slope Form Example continued: a. ) The year 1990 is represented by x = 0. 2005 is 15 years after 1990, so 2005 is represented by x = 15. The two points (0, 50) and (15, 85) will be used to find the equation. Substitute the values for m, x 1, and y 1. Distribute. Add 50 to both sides. Martin-Gay, Beginning and Intermediate Algebra, 4 ed Continued. 10

Using the Point-Slope Form Example continued: c. ) Use the equation employees in 2000. to predict the number of In 2000, x = 10. Martin-Gay, Beginning and Intermediate Algebra, 4 ed 11