1. ALGEBRA TWO Section 2.4 - Writing Equations of Lines
CHAPTER TWO: LINEAR EQUATIONS AND FUNCTIONS
Section Writing Equations of Lines

2. LEARNING GOALS Goal One - Write linear equations.
Goal Two - Write direct variation equations.

3. Writing an Equation of a Line
SLOPE - INTERCEPT FORM: Given the slope m and the y-intercept b, use the equation
y = mx + b.
POINT - SLOPE FORM: Given the slope m and a point (x1, y1), use the equation y - y1 = m (x - x1).
TWO POINTS: Given two points (x1, y1) and (x2, y2), use the formula m = (y2 - y1) / (x2 - x1) to find the slope m. Then use the point-slope form with this slope and either of the given points to write an equation of the line. 1

4. PROBLEM: Write an equation of the line that has m = -2/3 and b = -2
Writing an Equation Given the Slope and y-intercept
PROBLEM: Write an equation of the line that has m = -2/3 and b = -2
SOLUTION
y = mx + b
Use the slope-intercept form.
y = (-2/3)x - 2
Substitute (-2/3) for m and (-2) for b.
An equation of the line is y = (-2/3)x - 2

5. PROBLEM: Write an equation of the line that has m = 3 and b = 0
Writing an Equation Given the Slope and y-intercept
PROBLEM: Write an equation of the line that has m = 3 and b = 0
SOLUTION
y = mx + b
Use the slope-intercept form.
y = (3)x + 0
Substitute (3) for m and (0) for b.
An equation of the line is y = 3x

6. PROBLEM: Write an equation of the line that has m = 3/4 and b = 2
Writing an Equation Given the Slope and y-intercept
PROBLEM: Write an equation of the line that has m = 3/4 and b = 2
SOLUTION
y = mx + b
Use the slope-intercept form.
y = (3/4)x + 2
Substitute (3/4) for m and (2) for b.
An equation of the line is y = (3/4)x + 2

7. PROBLEM: Write an equation of the line that has m = -2 and b = -3
Writing an Equation Given the Slope and y-intercept
PROBLEM: Write an equation of the line that has m = -2 and b = -3
SOLUTION
y = mx + b
Use the slope-intercept form.
y = (-2)x + (-3)
Substitute (-2) for m and (-3) for b.
An equation of the line is y = -2x - 3

8. Writing an Equation Given the Slope and a Point
PROBLEM: Write an equation of the line that passes through (2, -1) and has a slope of -5.
SOLUTION
y - y1 = m(x - x1)
Use the point-slope form.
y - (-1) = (-5)(x - (2))
Substitute (-5) for m and (2) for x1 and (-1) for y1.
y + 1 = -5x + 10
Use distributive property
y = -5x + 9
Simplify
The equation for the line is y = -5x + 9

9. Writing an Equation Given the Slope and a Point
PROBLEM: Write an equation of the line that passes through (-3, -2) and has a slope of (1/3).
SOLUTION
y - y1 = m(x - x1)
Use the point-slope form.
y - (-2) = (1/3)(x - (-3))
Substitute (1/3) for m and (-3) for x1 and (-2) for y1.
y + 2 = (1/3)x + (1)
Use distributive property
y = (1/3)x - 1
Simplify
The equation for the line is y = (1/3)x - 1

10. Writing an Equation Given Two Points The slope for the line is m = 7/3
PROBLEM: Write an equation of the line that passes through (-1, -3) and (2, 4).
SOLUTION
m = (y2 - y1) /(x2 - x1)
Write formula for slope.
m = (4 - (-3))(2 - (-1))
Substitute.
m = (4 + 3) / (2 + 1)
Simplify
m = 7/3
Simplify
The slope for the line is m = 7/3

11. Writing an Equation Given Two Points
PROBLEM: Write an equation of the line that passes through (-1, -3) and (2, 4).
SOLUTION
y - y1 = m(x - x1)
Use the point-slope form.
y - (-3) = (7/3)(x - (-1))
Substitute (7/3) for m and (-1) for x1 and (-3) for y1.
y + 3 = (7/3)x + 7/3
Use distributive property
y = (7/3)x - (2/3)
Simplify
The equation for the line is y = (7/3)x – (2/3)

12. Writing an Equation Given Two Points The slope for the line is m = -3
PROBLEM: Write an equation of the line that passes through (2, 5) and (4, -1).
SOLUTION
m = (y2 - y1) /(x2 - x1)
Write formula for slope.
m = (-1 - (5))(4 - (2))
Substitute.
m = (-6) / (2)
Simplify
m = -3
Simplify
The slope for the line is m = -3

13. Writing an Equation Given Two Points
PROBLEM: Write an equation of the line that passes through (2, 5) and (4, -1).
SOLUTION
y - y1 = m(x - x1)
Use the point-slope form.
y - (5) = (-3)(x - (2))
Substitute (-3) for m and (2) for x1 and (5) for y1.
y - 5 = (-3)x + (6)
Use distributive property
y = (-3)x + 11
Simplify
The equation for the line is y = -3x + 11

14. Writing an Equation Given Two Points The slope for the line is m = 1
PROBLEM: Write an equation of the line that passes through (-2, 1) and (4, 7).
SOLUTION
m = (y2 - y1) /(x2 - x1)
Write formula for slope.
m = (7 - (1))(4 - (-2))
Substitute.
m = (6) / (4 + 2)
Simplify
m = 6/6 = 1
Simplify
The slope for the line is m = 1

15. Writing an Equation Given Two Points
PROBLEM: Write an equation of the line that passes through (-2, 1) and (4, 7).
SOLUTION
y - y1 = m(x - x1)
Use the point-slope form.
y - (1) = (1)(x - (-2))
Substitute (1) for m and (-2) for x1 and (1) for y1.
y - 1 = x + (2)
Use distributive property
y = x + 3
Simplify
The equation for the line is y = x + 3

16. VOCABULARY
Two variables x and y show direct variation provided y = kx and k =/= 0.
The nonzero constant k is called the constant of variation. 1

17. Writing and Using a Direct Variation Equation
PROBLEM: The variables x and y vary directly, and y = 3 when x = -4. Write an equation that relates the variables. Then find y when x = 4.
SOLUTION
y = kx
Write direct variation equation.
(3) = k(-4)
Substitute (-4) for x and (3) for y.
-3/4 = k
Solve for k.
The value of k = (-3/4). The equation is y = (-3/4)x

18. Writing and Using a Direct Variation Equation
PROBLEM: The variables x and y vary directly, and y = 3 when x = -4. Write an equation that relates the variables. Then find y when x = 4.
SOLUTION
y = (-3/4)x
Write direct variation equation.
y = (-3/4)(4)
Substitute (4) for x.
y = -3
Solve for y.
The value of y = -3 when x = 4.

19. Writing and Using a Direct Variation Equation
PROBLEM: The variables x and y vary directly, and x = 10 when y = Write an equation that relates the variables. Then find y when x = - 2.
SOLUTION
y = kx
Write direct variation equation.
(100) = k(10)
Substitute (10) for x and (100) for y.
10 = k
Solve for k.
The value of k = 10. The equation is y = 10x

20. The value of y = -20 when x = -2.
Writing and Using a Direct Variation Equation
PROBLEM: The variables x and y vary directly, and x = 10 when y = Write an equation that relates the variables. Then find y when x = - 2.
SOLUTION
y = 10x
Write direct variation equation.
y = 10(-2)
Substitute (-2) for x.
y = -20
Solve for y.
The value of y = -20 when x = -2.

21. Writing and Using a Direct Variation Equation
PROBLEM: The variables x and y vary directly, and x = -3 when y = 12. Write an equation that relates the variables. Then find y when x = -2.
SOLUTION
y = kx
Write direct variation equation.
(12) = k(-3)
Substitute (-3) for x and (12) for y.
-4 = k
Solve for k.
The value of k = -4. The equation is y = -4x

22. Writing and Using a Direct Variation Equation
PROBLEM: The variables x and y vary directly, and x = -3 when y = 12. Write an equation that relates the variables. Then find y when x = -2.
SOLUTION
y = -4x
Write direct variation equation.
y = -4(-2)
Substitute (-2) for x.
y = 8
Solve for y.
The value of y = 8 when x = -2.

23. ASSIGNMENT
pg #13, #17, #19, #23, #25, #27, #29, #33, #35, & #39.