1. WARM UP 1. Determine whether the point (2, 7) is on the line y = 3x + 1. 1. Find the slope of the line containing the points (4, 2) and (7, 5). 2.Find the equation of the line with slope 3 and containing the point (2, 5) (7) = 3(2) + 1 7 = 6 + 1 7 = 7 YES, IT IS. SLOPE = 1 (y – 5) = 3(x – 2) y – 5 = 3x - 6 y = 3x - 1

2. MORE EQUATIONS OF LINES

3. OBJECTIVES  Find the two-point equation of a line  Find the slope and y-intercept of a line from the slope- intercept equation  Graph equations in slope-intercept form  Find the standard form of a linear equation  Use linear equations for understanding problems.

4. VOCABULARY  Slope-intercept equation  Standard form  Two-point equation

5. TWO-POINT EQUATIONS OF LINES  Given two points, we can find an equation of a line containing them.  If we find the slope of a line by dividing the change in y by the change in x and substitute this value for m in the the point-slope equation, we obtain a two-point equation.  THEOREM 3-6: The Two-Point Equation – Any nonvertical line containing the points has an equation:

6. EXAMPLE  Find an equation of the line containing the points (2, 3) and (1, -4) We find the slope and then substitute in the two-point equation. We take (2, 3) as and (1, -4) as substituting y – 3 = 7(x – 2) simplifying y – 3 = 7x – 14 y = 7x – 11

7. EXAMPLE  In the previous example, we could have taken (1, -4) as and (2, 3) as and arrived at the same equation substituting simplifying y + 4 = 7x – 7 y = 7x – 11

8. TRY THIS…  Find an equation of the line containing the following pairs of points: a. (1, 4) and (3, -2)b. (3, -6) and (0, 4) y = -3x + 7y = - x + 4

9. SLOPE-INTERCEPT EQUATIONS OF LINES  If we know the slope and y-intercept, we can find an equation for the line. y – (-2) = 4(x -0) y + 2 = 4x y = 4x – 2  Suppose a line has slope -4 and y-intercept -2. From the point-slope equation we have: This is the slope-intercept equation of the line.

10. THEOREM 3-7 THE SLOPE-INTERCEPT EQUATION A nonvertical line with slope m and y-intercept b has an equation of y = mx + b

11. EXAMPLE  From any equation for a nonvertical line, we can find the slope-intercept equation by solving for y. There is no slope intercept equation for a vertical line because the line has not slope.  Find the slope and y-intercept of the line whose equation is y = 2x -3: y = 2x – 3 Slope 2 y-intercept -3

12. MORE EXAMPLES  Find the slope and y-intercept of the line whose equation is 3x – 6y – 7 = 0: -6y = -3x + 7 First solve for y. This puts the equation into slope-intercept form. The slope is, the y-intercept is.

13. TRY THIS…  Find the slope and y-intercept of each line: a.y = -5x + b.-2x + 3y – 6 = 0 c.2y – 6 = 0 m = -5 b= m = b = 2 m = b = 3

14. GRAPHING USING SLOPE-INTERCEPT FORM Graph 5y – 20 = -3x We plot (0, 4) and then find another point by moving 5 units to the right and 3 units down. The point has coordinates (5, 1). Solving for y, we find the slope-intercept form y = - + 4. Thus the y-intercept is 4 and the slope is -.

15. TRY THIS… Graph each equation using slope-intercept form. a.y = -3x – 1 b.7y = -4x - 21 c.6x = -5y -

16. FINDING THE STANDARD FORM  Any linear equation can be written so that 0 is on the right side. This is called standard form of a linear equation  Definition: The standard form of a linear equation is Ax + By + C = 0, where A and B are not both zeros.  In standard form, A, B, and C represent constants.  We can change the standard form of a linear equation to the slope-intercept equation by solving for y. This leads to the following theorem.

17. THEOREM 3-8 THE SLOPE OF A LINE WHOSE EQUATION IS: Ax + Bx = C is if B ≠ 0

18. EXAMPLE  Find the standard form and slope of the equation 7x = ¼ - 5y Using the addition property principle, adding 5y – ¼. This equation is of the form Ax + By + C = 0, where A = 7 and B = 5 and C = -1/4, the slope is –A/B = -7/5.