§ 1.5 Equations of Lines

Slope-Intercept Form of a Line 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 = 3x – 4 The slope is 3. The y-intercept is (0, -4).

Slope-Intercept Form Example: Find the slope and y-intercept of the line –3x + y = –5. First, we need to solve the linear equation for y. By adding 3x to both sides, y = 3x – 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)

Slope-Intercept Form Example: Find the slope and y-intercept of the line 2x – 6y = 12. First, we need to solve the linear equation for y. – 6y = – 2x + 12 Subtract 2x 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)

The Point-Slope Form Point-Slope Form of the Equation of a Line The point-slope form of the equation of a line is where m is the slope of the line and (x1, y1) is a point on the line. slope (x1, y1) point on the line

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, x1 = 4, y1 = 3 y – y1 = m(x – x1) y – (– 3) = 5(x – 4) Substitute the values for m, x1, and y1. y + 3 = 5x – 20 Simplify and distribute. y = 5x – 23 Subtract 3 from both sides.

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.

Finding the Equation Given Two Points Example continued: Distribute. Add 1 to each side. Simplify.

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.

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, x1, and y1. Distribute. Add 50 to both sides. Continued.

Using the Point-Slope Form Example continued: c.) Use the equation to predict the number of employees in 2000. In 2000, x = 10.