1. Review
The Coordinate Plane
Everything you want need to
know about writing, graphing, &
solving equations of lines.
Includes Bonus Content:
Your ultimate guide to Parallel &
Perpendicular Lines
2. EXAMPLE 1
Use slope and y-intercept to write an equation
Write an equation of the line with a slope of –2 and
a y-intercept of 5.
y = mx + b
Write slope-intercept form.
y = –2x + 5
Substitute –2 for m and 5 for b.
3. EXAMPLE 2
Standardized Test Practice
Which equation represents the line shown?
A
y=–
2
x+3
5
C
y=–
2
x+1
5
The slope of the line is
B
D
y=–
5
x+3
2
2
5
y = 3x +
rise
run
=
–2
= –
5
2
.
5
The line crosses the y-axis at (0, 3). So, the
y-intercept is 3.
y = mx + b
2
y=– 5 x+3
Write slope-intercept form.
2
Substitute – 5 for m and 3 for b.
5. GUIDED PRACTICE
for Examples 1 and 2
Write an equation of the line with the given slope and y-intercept.
1. Slope is 8; y-intercept is –7.
ANSWER
y = 8x – 7
6. GUIDED PRACTICE
for Examples 1 and 2
Write an equation of the line with the given slope and y-intercept.
3
2. Slope is 4 ; y intercept is –3.
3
x–3
y=
ANSWER
4
7. EXAMPLE 3
Write an equation of a line given two points
Write an equation of the line shown.
8. EXAMPLE 3
Write an equation of a line given two points
SOLUTION
STEP 1
Calculate the slope.
m =
STEP 2
y2 – y1
x2 – x1
=
–1 – (–5)
3–0
=
4
3
Write an equation of the line. The line crosses the y-axis at (0, –
5). So, the y-intercept is –5.
y = mx + b
Write slope-intercept form.
y= 4 x–5
3
Substitute 4 for m and 5 for b.
3
9. Write a linear function
EXAMPLE 4
Write an equation for the linear function f with the values f(0) = 5 and
f(4) = 17.
SOLUTION
STEP 1
Write f(0) = 5 as (0, 5) and f (4) = 17 as (4, 17).
STEP 2
Calculate the slope of the line that
passes through (0, 5) and (4, 17).
m =
y2 – y1
x2 – x1
=
17 – 5
4–0
=
12
4
= 3
10. EXAMPLE 4
STEP 3
Write a linear function
Write an equation of the line. The line crosses the y-axis at (0,
5). So, the y-intercept is 5.
y = mx + b
Write slope-intercept form.
y = 3x + 5
Substitute 3 for m and 5 for b.
ANSWER
The function is f(x) = 3x + 5.
12. GUIDED PRACTICE
4.
for Examples 3 and 4
Write an equation for the linear function f with the given values.
f(0) = –2, f(8) = 4
ANSWER
y =3 x – 2
4
13. GUIDED PRACTICE
5.
for Examples 3 and 4
Write an equation for the linear function f with the given values.
f(–3) = 6, f(0) = 5
ANSWER
y=–1 x+5
3
14. EXAMPLE 1
Write an equation given the slope and a point
Write an equation of the line that passes through the point (–1, 3) and
has a slope of –4.
SOLUTION
STEP 1
Identify the slope. The slope is – 4.
STEP 2
Find the y-intercept. Substitute the slope and the
coordinates of the given point in y = mx + b. Solve for b.
y = mx + b
Write slope-intercept form.
3 = –4(–1) + b
Substitute –4 for m, –1 for x, and
3 for y.
15. EXAMPLE 1
Write an equation given the slope and a point
–1 = b
STEP 3
Solve for b.
Write an equation of the line.
y = mx + b
Write slope-intercept form.
y = –4x – 1
Substitute –4 for m and –1 for b.
16. for Example 1
GUIDED PRACTICE
Write an equation of the line that passes through the point (6, 3) and
has a slope of 2.
ANSWER
y = 2x – 9
17. EXAMPLE 2
Write an equation given two points
Write an equation of the line that passes through (–2, 5) and (2, –1).
SOLUTION
STEP 1
Calculate the slope.
m=
STEP 2
y2 – y1
x2 – x1
=
–1 – 5
2 – (–2)
–6
=
4
= –
3
2
Find the y-intercept. Use the slope and the point (–2, 5).
y = mx + b
Write slope-intercept form.
18. EXAMPLE 2
Write an equation given two points
5=–
3
(–2)
2
2=b
STEP 3
+b
Substitute – 3 for m, –2 for x,
and 5 for y. 2
Solve for b.
Write an equation of the line.
y = mx + b
y=–
3
x +2
2
Write slope-intercept form.
Substitute – 3 for m and 2 for b.
2
19. EXAMPLE 33
EXAMPLE
Standardized Test Practice
Which function has the values f(4) = 9 and f(–4) = –7?
A f (x) = 2x + 10
C
B
D
f (x) = 2x – 13
STEP 1
f (x) = 2x – 14
Calculate the slope. Write f (4) = 9 as (4, 9) and f (–4) = –7
as (–4, –7).
m=
STEP 2
f (x) = 2x + 1
y2 – y1
x2 – x1
=
–7 – 9
–4 – 4
= –16
–8
= 2
Find the y-intercept. Use the slope and the point (4, 9).
y = mx + b
Write slope-intercept form.
20. EXAMPLE 3
Standardized Test Practice
9 = 2(4) + b
1=b
STEP 3
Substitute 2 for m, 4 for x, and 9 for y.
Solve for b.
Write an equation for the function. Use function
notation.
f (x) = 2x + 1
ANSWER
The answer is B. A B C D
Substitute 2 for m and 1 for b.
21. for Examples 2 and 3
GUIDED PRACTICE
2. Write an equation of the line that passes through
(1, –2) and (–5, 4).
ANSWER
y = –x – 1
22. for Examples 2 and 3
GUIDED PRACTICE
3. Write an equation for the linear function with values
f(–2) = 10 and f(4) = –2?
ANSWER
y = –2x + 6
23. EXAMPLE 2
Graph an equation in point-slope form
Graph the equation y + 2 = 2 (x – 3).
3
SOLUTION
Because the equation is in point-slope form, you know
that the line has a slope of 2 and passes through the
3
point (3, –2).
Plot the point (3, –2). Find a second
point on the line using the slope.
Draw a line through both points.
24. EXAMPLE 2
Graph equation in 2
GUIDED PRACTICE an for Examplepoint-slope form
2.
Graph the equation y – 1 = –(x – 2).
ANSWER
25. EXAMPLE 1
Write an equation of a parallel line
Write an equation of the line that passes through (–3, –5) and is parallel to
the line y = 3x – 1.
SOLUTION
STEP 1
Identify the slope. The graph of the given equation has a slope of 3. So,
the parallel line through (–3, –5) has a slope of 3.
STEP 2
Find the y-intercept. Use the slope and the given point.
26. EXAMPLE 1
y = mx + b
–5 = 3(–3) + b
4=b
Write an equation of a parallel line
Write slope-intercept form.
Substitute 3 for m, 3 for x, and 5 for y.
Solve for b.
STEP 3
Write an equation. Use y = mx + b.
y = 3x + 4
Substitute 3 for m and 4 for b.
27. for Example 1
GUIDED PRACTICE
1.
Write an equation of the line that passes through
(–2, 11) and is parallel to the line y = –x + 5.
ANSWER
y = –x + 9
28. EXAMPLE 3
Determine whether lines are perpendicular
STATE FLAG
The Arizona state flag is shown in a
coordinate plane. Lines a and b appear to be
perpendicular. Are they?
Line a:
12y = –7x + 42
Line b:
11y = 16x – 52
SOLUTION
Find the slopes of the lines. Write the equations in slope-intercept
form.
29. Determine whether lines are perpendicular
EXAMPLE 3
Line a:
12y = –7x + 42
7
y=–
Line b:
12
x +
42
12
11y = 16x – 52
y=
16
11
x –
52
11
ANSWER
7
16
The slope of line a is –
. The slope of line b is 11
12
The two slopes are not negative reciprocals, so lines a and b are not
perpendicular.
30. EXAMPLE 4
Write an equation of a perpendicular line
Write an equation of the line that passes through
(4, –5) and is perpendicular to the line y = 2x + 3.
SOLUTION
STEP 1
Identify the slope. The graph of the given equation has a slope of 2.
Because the slopes of perpendicular lines are negative reciprocals, the
slope of the perpendicular line through (4, –5) is .
–1
2
31. Write an equation of a perpendicular line
EXAMPLE 4
STEP 2
Find the y-intercept. Use the slope and the
given point.
y = mx + b
–5 =
–3 =
STEP 3
Write slope-intercept form.
–1 (4) + b
2
b
Substitute – 1 for m, 4 for x, and
2
–5 for y.
Solve for b.
Write an equation.
y = mx + b
y= –
1
x – 3
2
Write slope-intercept form.
Substitute –
1
for m and –3 for b.
2
32. GUIDED PRACTICE
for Examples 3 and 4
3. Is line a perpendicular to line b? Justify your answer
Line a:
2y + x = –12
Line b:
using slopes.
2y = 3x – 8
ANSWER
1
3
No; the slope of line a is – 2
, the slope of line b is 2 . The slopes are not
negative reciprocals so the lines are not perpendicular.
33. GUIDED PRACTICE
for Examples 3 and 4
4. Write an equation of the line that passes through
(4, 3) and is perpendicular to the line y = 4x – 7.
ANSWER
y= – 1 x+4
4
34. EXAMPLE 2
Determine whether lines are parallel or perpendicular
Determine which lines, if any, are parallel or perpendicular.
Line a: y = 5x – 3
Line b:
x + 5y = 2
Line c:
–10y – 2x = 0
SOLUTION
Find the slopes of the lines.
Line a: The equation is in slope-intercept form.
The slope is 5.
Write the equations for lines b and c in slope-intercept form.
35. EXAMPLE 2
Line b:
Determine whether lines are parallel or perpendicular
x + 5y = 2
5y = – x + 2
y=
Line c:
–
1
x +
5
2
5
–10y – 2x = 0
–10y = 2x
y=
–
1
x
5
36. EXAMPLE 2
Determine whether lines are parallel or perpendicular
ANSWER
Lines b and c have slopes of – 1 , so they are
5
parallel. Line a has a slope of 5, the negative reciprocal
1
of – 5 , so it is perpendicular to lines b and c.
37. GUIDED PRACTICE
for Example 2
Determine which lines, if any, are parallel or perpendicular.
Line a: 2x + 6y = –3
Line b:
y = 3x – 8
Line c:
–1.5y + 4.5x = 6
ANSWER
parallel: b and c; perpendicular: a and b, a and c
38. EXAMPLE 5
Solve a multi-step problem
LIBRARY
Your class is taking a trip to the public library. You can travel in small and
large vans. A small van holds 8 people and a large van holds 12 people.
Your class could fill 15 small vans and 2 large vans.
a.
Write an equation in standard form that models the possible
combinations of small vans and large vans that your class
could fill.
b. Graph the equation from part (a).
c. List several possible combinations.
39. EXAMPLE 5
Solve a multi-step problem
SOLUTION
a. Write a verbal model. Then write an equation.
8
s
+
12
l
=
p
Because your class could fill 15 small vans and 2 large vans, use
(15, 2) as the s- and l-values to substitute in the equation 8s + 12l =
p to find the value of p.
8(15) + 12(2) = p
144 = p
Substitute 15 for s and 2 for l.
Simplify.
Substitute 144 for p in the equation 8s + 12l = p.
40. EXAMPLE 5
Solve a multi-step problem
ANSWER
The equation 8s + 12l = 144 models the possible combinations.
b.
Find the intercepts of the graph.
Substitute 0 for s.
8(0) + 12l = 144
l = 12
Substitute 0 for l.
8s + 12(0) = 144
s = 18
41. EXAMPLE 5
Solve a multi-step problem
Plot the points (0, 12) and (18, 0). Connect
them with a line segment. For this problem
only nonnegative whole-number values of s
and l make sense.
c.
The graph passes through (0, 12), (6, 8), (12, 4), and (18, 0). So, four
possible combinations are 0 small and 12 large, 6 small and 8 large,
12 small and 4 large, 18 small and 0 large.
42. EXAMPLE 5
for Example 5
GUIDED PRACTICESolve a multi-step problem
Solve a multi-step problem
EXAMPLE 5
7. WHAT IF? In Example 5, suppose that 8 students decide not to go on
the class trip. Write an equation that models the possible combinations of
small and large vans that your class could fill. List several possible
combinations.
ANSWER
8s + 12l = 136; 17 small, 0 large; 14 small, 2 large; 11 small, 4 large; 8
small, 6 large; 5 small, 8 large; 2 small, 10 large
