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Math Problems
Grade 8
Solve a system of equations using any method
Solve using augmented matrices.
\newline
x
=
−
8
x = -8
x
=
−
8
\newline
−
9
x
−
10
y
=
−
18
-9x - 10y = -18
−
9
x
−
10
y
=
−
18
\newline
(_____, _____)
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Solve using augmented matrices.
\newline
7
x
+
y
=
17
7x + y = 17
7
x
+
y
=
17
\newline
x
=
1
x = 1
x
=
1
\newline
(_____, _____)
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Solve using augmented matrices.
\newline
4
x
−
6
y
=
−
12
4x - 6y = -12
4
x
−
6
y
=
−
12
\newline
y
=
−
4
y = -4
y
=
−
4
\newline
(_____, _____)
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Solve using augmented matrices.
\newline
y
=
3
y = 3
y
=
3
\newline
–
6
x
−
4
y
=
6
–6x − 4y = 6
–6
x
−4
y
=
6
\newline
(_____, _____)
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Solve using augmented matrices.
\newline
−
5
x
−
7
y
=
1
-5x - 7y = 1
−
5
x
−
7
y
=
1
\newline
y
=
7
y = 7
y
=
7
\newline
(_____, _____)
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Solve using augmented matrices.
\newline
10
x
+
8
y
=
−
20
10x + 8y = -20
10
x
+
8
y
=
−
20
\newline
y
=
−
5
y = -5
y
=
−
5
\newline
(_____, _____)
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Solve using augmented matrices.
\newline
y
=
4
y = 4
y
=
4
\newline
4
x
−
8
y
=
8
4x − 8y = 8
4
x
−8
y
=
8
\newline
(_____, _____)
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Solve
(
x
+
y
−
1
)
d
y
−
(
x
−
y
+
2
)
d
x
=
0
(x+y-1) d y-(x-y+2) d x=0
(
x
+
y
−
1
)
d
y
−
(
x
−
y
+
2
)
d
x
=
0
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x
=
2
5
x
+
9
=
5
+
3
x
\begin{array}{l}x=2 \\ 5 x+9=5+3 x\end{array}
x
=
2
5
x
+
9
=
5
+
3
x
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Solve for
s
s
s
.
\newline
2
s
+
1
=
9
s
=
\begin{array}{l} 2 s+1=9 \\ s= \end{array}
2
s
+
1
=
9
s
=
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Solve
2
x
+
6
=
13
2x + 6 = 13
2
x
+
6
=
13
using a double number line model
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6
6
6
.
\newline
4
x
−
3
y
=
12
−
x
+
3
y
=
6
\begin{array}{l} 4 x-3 y=12 \\ -x+3 y=6 \end{array}
4
x
−
3
y
=
12
−
x
+
3
y
=
6
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(b)
10
−
2
y
=
4
(
1
−
y
)
10-2 y=4(1-y)
10
−
2
y
=
4
(
1
−
y
)
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Multiplying \& Dividing Rational Numbers
\newline
Application Practice
\newline
Solve each problem below, showing all work.
\newline
1
1
1
. Tia borrowed
$
4.55
\$ 4.55
$4.55
from her friend on
5
5
5
different days to buy lunch. If she makes
$
15
\$ 15
$15
/hour babysitting and babysits for
3
3
3
hours, how much money will Tia have after paying her friend back the money she owes?
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3
x
+
5
y
=
11
2
x
−
2
y
=
20.
\begin{array}{l}3 x+5 y=11 \\ 2 x-2 y=20 .\end{array}
3
x
+
5
y
=
11
2
x
−
2
y
=
20.
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2
x
+
3
x
+
4
x
=
72
2x + 3x + 4x = 72
2
x
+
3
x
+
4
x
=
72
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a
+
2
+
b
+
3
+
c
+
6
=
25
b
−
a
=
c
\begin{array}{l}a+2+b+3+c+6=25 \\ b-a=c\end{array}
a
+
2
+
b
+
3
+
c
+
6
=
25
b
−
a
=
c
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A
2
2
2
GEN
\newline
12
12
12
−
3
-3
−
3
Trigonometric Functions of
G
\mathrm{G}
G
Find the reference angle.
\newline
1
1
1
)
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2
x
−
y
=
7
2
x
+
7
y
=
31
\begin{array}{l} 2 x-y=7 \\ 2 x+7 y=31 \end{array}
2
x
−
y
=
7
2
x
+
7
y
=
31
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Use the equation below to find
A
A
A
, if
b
=
4
b=4
b
=
4
and
h
=
9
h=9
h
=
9
.
\newline
A
=
b
h
A = bh
A
=
bh
\newline
A
=
□
A = \square
A
=
□
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Solve the system of equations.
\newline
2
x
−
y
=
2
7
x
+
y
=
7
\begin{array}{l}2 x-y=2 \\ 7 x+y=7\end{array}
2
x
−
y
=
2
7
x
+
y
=
7
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10
10
10
.
{
13.8
x
+
9.2
y
=
−
4
6
x
+
4
y
=
8
\left\{\begin{array}{l}13.8 x+9.2 y=-4 \\ 6 x+4 y=8\end{array}\right.
{
13.8
x
+
9.2
y
=
−
4
6
x
+
4
y
=
8
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{
3
x
−
4
y
=
13
−
9
x
+
12
y
=
26
\left\{\begin{array}{l}3 x-4 y=13 \\ -9 x+12 y=26\end{array}\right.
{
3
x
−
4
y
=
13
−
9
x
+
12
y
=
26
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3
3
3
. Write two equivalent ratios.
\newline
1
1
1
)
\newline
\begin{tabular}{|l|l|l|}
\newline
\hline
2
2
2
&
4
4
4
& \\
\newline
\hline
6
6
6
&
12
12
12
& \\
\newline
\hline
\newline
\end{tabular}
\newline
2
2
2
)
\newline
\begin{tabular}{|l|l|l|}
\newline
\hline
1
1
1
&
3
3
3
& \\
\newline
\hline
4
4
4
& & \\
\newline
\hline
\newline
\end{tabular}
\newline
4
4
4
)
\newline
\begin{tabular}{|l|l|l|}
\newline
\hline
8
8
8
& &
32
32
32
\\
\newline
\hline
9
9
9
& &
36
36
36
\\
\newline
\hline
\newline
\end{tabular}
\newline
5
5
5
)
\newline
\begin{tabular}{|l|l|l|}
\newline
\hline
3
3
3
& & \\
\newline
\hline
7
7
7
& & \\
\newline
\hline
\newline
\end{tabular}
\newline
Page
12
12
12
of
13
13
13
\newline
Stage
1
1
1
Essential Mathematics - AT
1
1
1
- Topic
1
1
1
- Calculati
\newline
Adapted from the resources provided
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2
X
+
3
y
=
5
,
5
x
+
7
y
=
9
2X+3y=5,5x+7y=9
2
X
+
3
y
=
5
,
5
x
+
7
y
=
9
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Solve
\newline
w
+
3
4
=
2
\frac{w+3}{4}=2
4
w
+
3
=
2
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5
x
+
3
y
=
9
2
x
−
3
y
=
12
\begin{array}{l}5 x+3 y=9 \\ 2 x-3 y=12\end{array}
5
x
+
3
y
=
9
2
x
−
3
y
=
12
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3
>
u
+
3
3
3>\frac{u+3}{3}
3
>
3
u
+
3
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Solve the system of equations.
\newline
3
x
−
2
y
=
6
x
+
y
=
−
8
\begin{array}{}3 x-2 y=6 \\ x+y=-8\end{array}
3
x
−
2
y
=
6
x
+
y
=
−
8
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−
8
x
+
2
y
=
46
y
=
3
x
+
58
\begin{array}{c}-8 x+2 y=46 \\ y=3 x+58\end{array}
−
8
x
+
2
y
=
46
y
=
3
x
+
58
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4
x
−
7
y
=
15
−
9
x
+
y
=
−
19
\begin{array}{l}4 x-7 y=15 \\ -9 x+y=-19\end{array}
4
x
−
7
y
=
15
−
9
x
+
y
=
−
19
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Use matrix to solve the simultaneay equation
\newline
x
+
y
+
z
−
4
=
0
2
x
−
3
y
+
4
z
−
33
=
0
3
x
−
2
y
−
2
z
−
2
=
0
\begin{array}{l} x+y+z-4=0 \\ 2 x-3 y+4 z-33=0 \\ 3 x-2 y-2 z-2=0 \end{array}
x
+
y
+
z
−
4
=
0
2
x
−
3
y
+
4
z
−
33
=
0
3
x
−
2
y
−
2
z
−
2
=
0
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12
x
+
9
y
=
20
4
x
+
7
y
=
80
×
3
\begin{array}{l}12 x+9 y=20 \\ 4 x+7 y=80 \times 3\end{array}
12
x
+
9
y
=
20
4
x
+
7
y
=
80
×
3
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Write a system of equations to describe the situation below, solve using any method, and fill in the blanks.
\newline
The members of a cooking club are making cakes, which they will sell at a street fair for
$
23
\$23
$23
apiece. It cost
$
20
\$20
$20
for a booth at the fair, and the ingredients for each cake cost
$
3
\$3
$3
. At some point, the club members will sell enough cakes so that their sales cover their expenditures. How many cakes will they have sold? How much will the sales and expenditures be?
\newline
Once the members of the club have sold _____ cakes, they will break even, with sales and expenditures of
$
\$
$
_____.
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Write a system of equations to describe the situation below, solve using any method, and fill in the blanks.
\newline
Two students in Mr. Terry's class, Britney and Francesca, have been assigned a workbook to complete at their own pace. They get together at Britney's house after school to complete as many pages as they can. Britney has already completed
13
13
13
pages and will continue working at a rate of
11
11
11
pages per hour. Francesca has completed
27
27
27
pages and can work at a rate of
9
9
9
pages per hour. Eventually, the two students will be working on the same page. How long will that take? How many pages will each of them have completed?
\newline
After
_
\_
_
hours, Britney and Francesca will have each completed
_
\_
_
pages in their workbooks.
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7
(
2
e
−
1
)
−
3
=
6
+
6
e
e
=
\begin{array}{l}7(2 e-1)-3=6+6 e \\ e=\end{array}
7
(
2
e
−
1
)
−
3
=
6
+
6
e
e
=
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{
2
x
−
y
+
3
z
=
13
x
+
2
y
−
3
z
=
6
3
x
+
8
y
+
2
z
=
3
\left\{\begin{array}{c}2 x-y+3 z=13 \\ x+2 y-3 z=6 \\ 3 x+8 y+2 z=3\end{array}\right.
⎩
⎨
⎧
2
x
−
y
+
3
z
=
13
x
+
2
y
−
3
z
=
6
3
x
+
8
y
+
2
z
=
3
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{
x
+
2
y
+
z
=
−
2
x
+
2
y
−
2
z
=
−
5
−
x
+
y
+
3
z
=
12
\left\{\begin{array}{c}x+2 y+z=-2 \\ x+2 y-2 z=-5 \\ -x+y+3 z=12\end{array}\right.
⎩
⎨
⎧
x
+
2
y
+
z
=
−
2
x
+
2
y
−
2
z
=
−
5
−
x
+
y
+
3
z
=
12
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{
x
+
y
+
z
=
10
x
−
y
+
z
=
−
4
−
x
+
y
=
6
\left\{\begin{array}{c}x+y+z=10 \\ x-y+z=-4 \\ -x+y=6\end{array}\right.
⎩
⎨
⎧
x
+
y
+
z
=
10
x
−
y
+
z
=
−
4
−
x
+
y
=
6
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Which of the following is a rational number?
\newline
Choices:
\newline
(A)
π
\pi
π
\newline
(B)
1.27
1.27
1.27
\newline
(C)
7
\sqrt{7}
7
\newline
(D)
3
\sqrt{3}
3
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Which of the following is an irrational number?
\newline
Choices:
\newline
(A)
π
\pi
π
\newline
(B)
4
4
4
\newline
(C)
0
0
0
\newline
(D)
5
5
5
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Solve.
\newline
7
x
−
10
y
=
20
7x - 10y = 20
7
x
−
10
y
=
20
\newline
y
=
−
9
y = -9
y
=
−
9
\newline
(_____, _____)
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Solve.
\newline
8
x
+
3
y
=
−
11
8x + 3y = -11
8
x
+
3
y
=
−
11
\newline
y
=
−
9
y = -9
y
=
−
9
\newline
(_____, _____)
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Solve.
\newline
8
x
−
8
y
=
16
8x - 8y = 16
8
x
−
8
y
=
16
\newline
x
=
8
x = 8
x
=
8
\newline
(_____, _____)
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Solve.
\newline
5
x
+
6
y
=
20
5x + 6y = 20
5
x
+
6
y
=
20
\newline
x
=
−
8
x = -8
x
=
−
8
\newline
(_____, _____)
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Solve.
\newline
x
=
6
x = 6
x
=
6
\newline
6
x
+
9
y
=
18
6x + 9y = 18
6
x
+
9
y
=
18
\newline
(_____, _____)
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{
2
x
−
4
y
+
5
z
=
−
40
6
x
+
3
y
−
2
z
=
−
5
−
4
x
−
y
+
3
z
=
−
3
⟩
B
\left\{\begin{array}{l}2 x-4 y+5 z=-40 \\ 6 x+3 y-2 z=-5 \\ -4 x-y+3 z=-3\end{array}\right\rangle B
⎩
⎨
⎧
2
x
−
4
y
+
5
z
=
−
40
6
x
+
3
y
−
2
z
=
−
5
−
4
x
−
y
+
3
z
=
−
3
⟩
B
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{
w
−
x
+
y
+
z
=
5
3
w
−
2
x
−
5
y
−
3
z
=
8
3
w
+
x
−
y
−
z
=
11
−
w
+
2
x
+
3
y
+
z
=
−
4
\left\{\begin{array}{rr}w-x+y+z= & 5 \\ 3 w-2 x-5 y-3 z= & 8 \\ 3 w+x-y-z= & 11 \\ -w+2 x+3 y+z= & -4\end{array}\right.
⎩
⎨
⎧
w
−
x
+
y
+
z
=
3
w
−
2
x
−
5
y
−
3
z
=
3
w
+
x
−
y
−
z
=
−
w
+
2
x
+
3
y
+
z
=
5
8
11
−
4
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The principal's new car cost
$
35
,
000
\$ 35,000
$35
,
000
, but in three years it will only be worth
$
21
,
494
\$ 21,494
$21
,
494
. Write an equation for this situation and explain what each part of the equation represents in the context of the problem. What is the annual rate of depreciation?
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4
x
+
8
y
=
20
4x+8y=20
4
x
+
8
y
=
20
\newline
−
4
x
+
2
y
=
−
30
-4x+2y=-30
−
4
x
+
2
y
=
−
30
\newline
Consider the given system of equations. How many
(
x
,
y
)
(x,y)
(
x
,
y
)
solutions does this system have?
\newline
Choose
1
1
1
answer:
\newline
(A) No solutions
\newline
(B) Exactly one solution
\newline
(C) Infinitely many solutions
\newline
(D) None of the above
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