Date
Name
Slope-intercept form of linear equations
WriHng Unear Equations
linear equations are equations of lines. A linear equation written in slope-intercept form is
mx + b, where m is the slope of the line and b is the y-intercept. The y-intercept is the
point at which the Linecrosses the y-axis.
y = lX + 2
Y = -SX + 1
y =x - 7
Y :::I -8x - 10
3, y-int. = 2
m = -5, y-int. ;::1
m;:: 1,y-int. ;::-7
m;:: -8, y-int. = -10
y =
, m:::l
Equations written in standard form can be put in slope-intercept form simply by adding or
sUbtracting terms from either side of the equations.
3X + Y = 12
-7 X - Y ;::14
6X - 2y = 10
lX - 3x + Y = 12 - lX
-7X + 7X - Y = 14 + 7X
6X - 6x - 2Y = 10- 6X
Y = -JX + 12
-y = 7X + 14
~=.l!._~
(-l)-y
= (-1)(7X
+
-2
II)
y=-7X-1I
1. Write the slope-intercept
-2
Y
= -5
Y
=
+
-2
3x
lX - 5
form of the equation of a line.
State the slope and the y-intercept of each line.
2. Y= 2x+ 5
3. Y= x- 10
4. Y= -x+ 4
5. Y= -3x+ 7
6. Y= -5x
7. y=5
Put each equation In slope-intercept
8. 3x+ Y= 10
form.
9. 7x- Y= -12
= 12 + 3x
10. -7x- y=-5
11. -3Y
12. 4y= 4x+ 8
13. x+ 2y= 16
4. -2x+ 8y=-8
15. - 12Y = 24x + 12
:arson-Dellosa
CD-4324
Algebra
Date
Name
WrlHng Linear EquaHons
Linear equations using slope-intercept form
Remember,the slope-intercept form of a linear equation is Ii = mx + b, where m is the Slope
and b is the v-intercept. Towrite an equation of a line in this form, you would need the slope
and the v-intercept of the line.
m = 2, y-int. = 4
m = -4, y-int. = -q
m = -I, y-int. = "
y • mx + b
Y = mx + b
Y = mx + b
Ii
= 2X + 4
Y = -4x - q
Y •• -x
+ "
note: In the third example, the slope is -1. Only -x is written. If the slope was I, only x would
I is simPly understood to be there and usuaHy is not written.
be written. The coefficient
~I
State the slope and y-Intercept of each line below.
1.
v=
-2x+ 5
2. y= -8x
3. 2 + y=-x
4. -9+ y= x
5. y= 7
6.
7. -6 + v= 4x
8. y=-x
v=
2x- 7
", • , , "'7
Write an equation of the line given its slope and y-Intercept.
9. m
= 4, y~int. = ~1
10. m
11. m
= -2, v-into = -5
12. m e
13. m
1
= 2'
j-lnt,
14. m = -1, y~lnt.= 8
=-6
15. m = 1, y-Int. = 2
16.
= 0, y-Int. = 7
:
!. j-lnt. =.3
m = 4, y-Int.
5
=0
17. What Isthe name of the y = mx + b form of an equation of a line?
8. Is 7 the x-Intercept or y-Intercept of the line y
Carson-Dellaso
CD-4324
= -4x + 7?
31
Algebra
Date
Name
Linear equations given the slope and a point
WrlHng Unear Equations
Remember the slope-intercept form of an equation is y •• mx + b, where m is the slope and
is the v-intercePt. This form can be helpful in finding the equation of a line given the
Slope and any point on the line. The point (x, y) is just substituted in for x and y in the
equation. Also SUbstituting the value of m in for the slope, you are left with b, which is the
y-intercept. Oncethe slope and the y-intercept are found, the equation of the given line
can be written.
•
b
point (5, If),
m .• 2
point (-2, 3), m = 14
Y •• mx + by
If = 2(5) + b
If = fO + b
-6 ••b
•• mx + b
l •• 11(-2) + b
3 •• -8 + b
rr = b
Solve each equation for b.
1
1. -3 = 2(-4)
+b
4. -5 = 7(-2) + b
5. 6 = (7)(-1) + b
6. 7 = -9(-2) + b
Given the point and slope of each line, find the y-Intercept.
7. (-4, 7), m = 3
10. (-3, -5),
m =-2
8. (0, 1), m
=4
11. (4, 3), m
= -3
9. (-4, -2), m
= -1
12. (3, 1), m = -3
I
(
I
I
State the equation of each line using Its slope and y-Intercept In problems 7-12. Write the
equation In slope-Intercept form.
13.
14.
15.
16.
17.
18.
Write the slope-Intercept
has the given slope.
19. m
= ~,
point (3,5)
© Carson-Dellose CD-4324
I
\
I
I
I
I
form of the equation of a line that passes through the point and
20. m
32
= -2, point
(-2,6)
Algebra
I
I
Date
Name
Linear equations given two points
WrlHng Unear EquaHons
Tofind any equation of a line, the slope of the line must be found. When just two points are
given and the equation of the Linethat contains these points must be found, the first step to
take is to find the slope of the line. The equation for the slope, m, of a line given two points is
m
=
~~~
=
I}z - I},
,
xz - x,
where (x" Y,), and (xz, Yz) are the
given points on the same line.
points (5, -7) and (7, 5)
points (2, 0) and (-If, -3)
m=
m.
rise = 5 - (-7) = .!! = 6
run
7 _ 5
m=6
nOW that
equation
example
equation
rise = -3 - 0 =.:!
run
2
m="2
-II - 2
= .l.
-6
2
I
the slopes are found given the sets of points above. move ahead and find the
of each tine. Chooseone point from each example and use the slope found in its
to find the y-intercept. Once the y-intercept is found, you have everything to write the
of each tine.
point (7,5), m os 6
point (2, 0), m =
y s mx + b
y = mx + b
+
5 •• 6(7)
+
0= .l.(2)+ b
b
2
O=I+b
-I = b
5 '" 112 + b
-37'" b
The equation of the tine is y • 6X - 37.
+
The equation of the line is y ••
x-I.
Find the slope of each line given each set of points.
1. (-1, -1), (2, 8)
2. (3, 1), (-3, 5)
3. (6, -5), (1, 5)
4. (1, 2), (2, 4)
S. (7, 4), (-2, -5)
6. (1, 8), (8, 8)
Write the slope-Intercept
polnts above.
form of the equation of each line that passes through the given
7.
8.
9.
10.
11.
12.
Write the Slope-Intercept form of the equation of the line that passes through each set of points.
13. (-3, -3), (-6, 0)
© Carson-Dellosa CD-4324
14. (1, 1), (-3, 1)
33
15. (-1, 4), (-3, 8)
Algebra
Date
Name
Slope of a line
Graphing Linear Equations
The slope of a line is the number of units a line rises or falls for each unit of horizontal
change from left to right on its graph. The slope of a line is represented by the letter m
and can be found given two points (x" Y,)and (xz' Yz) on the line using the following formula.
m
If
If
If
If
the
the
the
the
=
rise
run
=
Vz - Y,
- x,
Xz
slope is greater than 0 (which is positive), then the line rises from left to right.
slope is less than -0 (which isnegative), then the line taus from left to right.
slope is equal to 0, then the tine is horizontal.
slope is undefined, then the tine is vertical.
Find the slope of the line passing through the points (-5, 6) and (-3, 10).
m = rise =
run
V2 - V,
Xz
-
X,
=
10- 6
= ~
1#
=
-3 - (-5)
-3 +
5
= .! = 2
2
I
Thus, since the slope is a positive 2, then we know the line rises from left to right.
Determine whether the line rises trom left to right, falls from left to right, Isvertical, or is horizontal,
given the fo"owlng slopes.
1. m=2
2. m=-5
3. m=6
4. m = undefined
5. m=O
6. m =-7
Find the slope of the line that passes through each set of points.
7. (5, 4), (6, 9)
8. (-3, 4), (-1, 2)
9. (-6, -3), (-2, 9)
10. (-2, -1), (0, 3)
11. (6, -5), (3, 10)
12. (7,8), (1. -16)
Sketch the line by plotting the points. Find the slope of the line passing through the points.
13. (0, 0), (2, 4)
© Corson-Dellosa CD-4324
14. (-1. -2), (-3, 6)
15. (4, -5), (5, -10)
"7
16. (-2,0), (-5, 6)
Algebra
Date
Name
Graphing using slope-intercept form
Graphing Unear Equatton.
Tograph a line using its stope and y-intercept, the first thing to do is put the equation in its
slope-intercept form, u .. mx + b. Then m is the stene of the line and b is the y-intercept. Here
are the steps to graph a Lineusing its slope and y-intercept.
1. Put the Lineequation in sLope-intercept form.
2. Find the slope and v-intercept.
3. Graph the v-intercept.
14. Graph the slope. start at the y-intercept point, move up or down, then right or left (rise
over the run).
Graph the Line 5X + y •• 7.
ya -5X+ 7
Put the line in slope-intercept form.
Find the slope and y-intercept.
m • -5, v-intercept .• 7
now, graph (0, 7). Move from this point
down 5 places and then to the right 1,
since the slope is The graph answer point is (t, 2).
Thus, you have the graph of the line 5X + Y = 7.
t.
Solve each equation for y.
1. 7x-4y=0
2. y+ 18 = 0
3. 3x- 2y= -12
4. -8x+ v= 4
5. -4x- 5y= 40
6. Tx « 2y= -10
Find the slope and y-Intercept of each line.
7. y= -3x+ 5
8. y= 7x+ 11
Sketch the two lines on the same coordinate
of each line.
v=
14.
15. y = 3x + 3, y = -3x - 3
CD-4324
3x- 4
plane. State the slope, x-Intercept, and y-Intercept
-2x+ 4, y= -2x- 3
e Carson-Dellosa
X=
12. x- y=-9
11. x+y=2x-5
10. y=-2+4x
13.
9. 2y-
v=
x+ 1, y= -x+ 1
16. Y= 5x, Y= -5x
If.
Algebra
Date
Name
I
Graphing Unear Equallonl
Unit 5 Test
i
I
I
I
1. Sketch the graphs of x = -2 and y
2. Solve 3x + 4 y
its graph.
= -24 for
= 5. Then state the
I
point at which these two lines Intersect.
y. Give three solutions to this equation and use these points to sketch
t
r
I
I
I
I
t
3. Write an equation of a line whose x-coordinates are all 7. Give three points on this line.
~
~
(
I
4. Is (4, -2) on the graph of -3x - y = 10? Explain why or why not.
(
Find the x-Intercept and v-intercept in each equation. Sketch the line.
5.
v=
-6x- 3
6. 5x- 2y=-1O
7. 2x- y=-8
Sketch each line which passes through the two points given. State the slope of the line and
whether the line rises 'rom right to left, falls from right to left, is vertical, or Ishorizontal.
8. (-2, 4), (2, 4)
9. (-6, -3), (-3, 9)
Rewrite each equation In slope-Intercept
Sketch the graph.
11. -7x+y=-1
10. (7, -9), (7, 12)
form. State the slope and V-intercept of each line.
12.2x-2y=4x-1O
13. 4(2x+y)=
16
Graph the following equations. State the method you chose to use and explain why.
14. 7x - 8y = -56
© Carson-Dellosa CD-4324
15. y = 2x - 6
16. -x - 7y = 14
so
Algebra
Date
Name
~.
i'
Working with Inequalities
and Absolute Value.
Graphing linear inequalities with two variables
A linear inequality can be written in the fallowing forms:
ax
+
by
<
c
ax
+
by
ax
1C
+
(
by ) C
ax
+
by ~ c
l
where (x, !/) is an ordered pair that is a solution of the linear inequality, making the
inequality true.
Is (-2, 5) a solution of zx - 6y ) 127
2(-2) - 6(5) ) 12
Substitute the x and !/ values into the inequality.
(
t
•
t
-II - 30 ) 12
-311 ) 12
•
not true. Therefore, (-2, 5) is not a solution.
Tosketch a graph of a linear inequality, follow these simple steps.
J. Sketch the graph of the corresponding linear equation using a dashed line for < or )
and a solid line for 1 or ~.Thus, separating the coordinate plane into two half planes.
2. Pick a point in each of the half planes and test each to find which one is a solution to
the linear ineqUality.
3. Shade the half of the plane that contains the point that is a solution to
the linear ineqUality.
The graph of the above example of zx - 6Y) 12is shown.
note: Since (-2, 5) was not a solution, choose another point on the other
half of the plane to show what half needs to be shaded.
1
1
I
II
xt++-t+++t+++
I
Write ye. or no to state whether the given point Isa solution of the mequolltv.
1. 2x- 3ys 7; (5, -4)
2. -x-y>5:(-2,-4)
3. 5x + 4y~ 8: (-2, 6)
4. -7x+ 8y< 12: (-3, 2)
Sketch the graph of each Inequa"ty.
5. x+ y< 5
6. x>-1
7. 2x- y~ 2
8.
9. -3x+ 4ys 12
© Corson-Deaoso CD-4324
vs
a
10. 5x - 2 y > - 10
611
f
Algebra