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Math Problems
Grade 8
Write and solve direct variation equations
Evaluate
0.3
y
+
y
z
0.3y + \frac{y}{z}
0.3
y
+
z
y
when
y
=
10
y = 10
y
=
10
and
z
=
5
z = 5
z
=
5
.
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3
+
j
k
+
k
3
3 + jk + k^3
3
+
jk
+
k
3
when
j
=
2
j = 2
j
=
2
and
k
=
6
k = 6
k
=
6
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7
p
=
9
(
p
+
q
)
+
11
7p = 9(p + q) + 11
7
p
=
9
(
p
+
q
)
+
11
\newline
99
+
3
=
4
(
7
q
+
p
)
99 + 3 = 4(7q +p)
99
+
3
=
4
(
7
q
+
p
)
\newline
Consider the system of equations. If
p
,
9
p, 9
p
,
9
is the solution to the system, then what is the value of
−
p
−
9
-p - 9
−
p
−
9
?
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Let
x
x
x
and
y
y
y
be functions of
t
t
t
with
y
=
4
3
π
x
3
y = \frac{4}{3} \pi x^3
y
=
3
4
π
x
3
. If
d
x
d
t
=
1
16
\frac{dx}{dt} = \frac{1}{16}
d
t
d
x
=
16
1
, what is
d
y
d
t
\frac{dy}{dt}
d
t
d
y
when
x
=
6
x = 6
x
=
6
?
\newline
Write an exact, simplified answer.
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If
A
=
{
a
,
b
,
c
,
d
,
e
}
A= \{a, b, c, d, e\}
A
=
{
a
,
b
,
c
,
d
,
e
}
and
B
=
{
c
,
d
,
f
,
g
}
B= \{c, d, f, g\}
B
=
{
c
,
d
,
f
,
g
}
, what is the set difference
B
/
A
B/A
B
/
A
?
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Given
k
(
x
)
=
{
2
x
2
−
4
k(x) = \{2x^2 - 4
k
(
x
)
=
{
2
x
2
−
4
for
x
<
2
x < 2
x
<
2
and
−
3
x
+
10
-3x + 10
−
3
x
+
10
for
x
x
x
is equal to or less than
2
2
2
. Is
k
(
x
)
k(x)
k
(
x
)
continuous at
x
=
2
x=2
x
=
2
? Explain. If
j
(
x
)
=
k
(
x
+
1
)
j(x)=k(x+1)
j
(
x
)
=
k
(
x
+
1
)
, what is the function for
j
(
x
)
j(x)
j
(
x
)
?
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Solve the surface area formula
A
=
6
s
2
A=6s^{2}
A
=
6
s
2
for
s
s
s
. Then find the value of
s
s
s
when
A
=
24
A=24
A
=
24
.
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A direct variation includes the points
(
−
20
,
−
40
)
(-20,-40)
(
−
20
,
−
40
)
and
(
7
,
n
)
(7,n)
(
7
,
n
)
. Find
n
n
n
. Write and solve a direct variation equation to find the answer.
\newline
n
=
_
_
_
n = \_\_\_
n
=
___
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If
y
y
y
varies directly with
x
x
x
and
y
=
−
26
y = -26
y
=
−
26
when
x
=
26
x = 26
x
=
26
, find
y
y
y
when
x
=
21
x = 21
x
=
21
. Write and solve a direct variation equation to find the answer.
\newline
y = ____
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A direct variation includes the points
(
6
,
90
)
(6,90)
(
6
,
90
)
and
(
2
,
n
)
(2,n)
(
2
,
n
)
. Find
n
n
n
. Write and solve a direct variation equation to find the answer.
\newline
n = ____
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y
=
1
3
x
+
5
y = \frac{1}{3}x+5
y
=
3
1
x
+
5
\newline
y
=
2
x
y= 2x
y
=
2
x
\newline
Consider the given system of equations. If
(
x
,
y
)
(x,y)
(
x
,
y
)
is the solution to the system, then what is the value of
\newline
y
+
x
y+x
y
+
x
?
\newline
◻
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If
f
(
1
)
=
9
f(1)=9
f
(
1
)
=
9
and
f
(
n
)
=
−
4
f
(
n
−
1
)
f(n)=-4 f(n-1)
f
(
n
)
=
−
4
f
(
n
−
1
)
then find the value of
f
(
4
)
f(4)
f
(
4
)
.
\newline
Answer:
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Solve the following equation for
x
x
x
. Express your answer in the simplest form.
\newline
4
(
7
x
+
5
)
=
28
x
+
21
4(7 x+5)=28 x+21
4
(
7
x
+
5
)
=
28
x
+
21
\newline
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Solve the following equation for
x
x
x
. Express your answer in the simplest form.
\newline
−
5
(
−
4
x
+
3
)
=
16
x
−
16
-5(-4 x+3)=16 x-16
−
5
(
−
4
x
+
3
)
=
16
x
−
16
\newline
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12
t
=
4
v
−
3
12t = 4v - 3
12
t
=
4
v
−
3
\newline
−
6
t
=
4
v
+
6
-6t = 4v + 6
−
6
t
=
4
v
+
6
\newline
If
(
t
,
v
)
(t, v)
(
t
,
v
)
is the solution to the system of equations, what is the value of
t
−
v
t - v
t
−
v
?
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Use the given information to find the unknown value:
y
y
y
varies directly as the cube of
x
x
x
. When
x
=
2
x=2
x
=
2
, then
y
=
16
y=16
y
=
16
. Find
y
y
y
when
x
=
3
x=3
x
=
3
.
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If
y
y
y
varies directly with
x
x
x
and
y
=
14
y = 14
y
=
14
when
x
=
2
x = 2
x
=
2
, find
y
y
y
when
x
=
1
x = 1
x
=
1
. Write and solve a direct variation equation to find the answer.
\newline
y
y
y
= ____
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A direct variation includes the points
(
3
,
15
)
(3,15)
(
3
,
15
)
and
(
1
,
n
)
(1,n)
(
1
,
n
)
. Find
n
n
n
. Write and solve a direct variation equation to find the answer.
\newline
n
=
_
_
_
n = \_\_\_
n
=
___
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If
y
y
y
varies directly with
x
x
x
and
y
=
15
y = 15
y
=
15
when
x
=
5
x = 5
x
=
5
, find
y
y
y
when
x
=
1
x = 1
x
=
1
. Write and solve a direct variation equation to find the answer.
\newline
y
y
y
= ____
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If
y
y
y
varies directly with
x
x
x
and
y
=
12
y = 12
y
=
12
when
x
=
4
x = 4
x
=
4
, find
y
y
y
when
x
=
1
x = 1
x
=
1
. Write and solve a direct variation equation to find the answer.
\newline
y
y
y
= ____
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If
y
y
y
varies directly with
x
x
x
and
y
=
8
y = 8
y
=
8
when
x
=
4
x = 4
x
=
4
, find
y
y
y
when
x
=
3
x = 3
x
=
3
. Write and solve a direct variation equation to find the answer.
\newline
y
y
y
= ___
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If
y
y
y
varies directly with
x
x
x
and
y
=
20
y = 20
y
=
20
when
x
=
10
x = 10
x
=
10
, find
y
y
y
when
x
=
4
x = 4
x
=
4
. Write and solve a direct variation equation to find the answer.
\newline
y
=
‾
y = \underline{\hspace{2em}}
y
=
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A direct variation includes the points
(
4
,
16
)
(4,16)
(
4
,
16
)
and
(
1
,
n
)
(1,n)
(
1
,
n
)
. Find
n
n
n
. Write and solve a direct variation equation to find the answer.
\newline
n
=
_
_
_
n = \_\_\_
n
=
___
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If
x
x
x
and
y
y
y
are in direct proportion and
y
y
y
is
36
36
36
when
x
x
x
is
9
9
9
, find
y
y
y
when
x
x
x
is
8
8
8
.
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If
x
x
x
and
y
y
y
are in direct proportion and
y
y
y
is
28
28
28
when
x
x
x
is
4
4
4
, find
y
y
y
when
x
x
x
is
3
3
3
.
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Find the mean of the
32
32
32
numbers, such that if the mean of
10
10
10
of them is
15
15
15
and the mean of
20
20
20
of them is
11
11
11
. The last two numbers are
10
10
10
.
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If
a
1
=
3
,
a
2
=
1
a_{1}=3, a_{2}=1
a
1
=
3
,
a
2
=
1
and
a
n
=
3
a
n
−
1
−
2
a
n
−
2
a_{n}=3 a_{n-1}-2 a_{n-2}
a
n
=
3
a
n
−
1
−
2
a
n
−
2
then find the value of
a
5
a_{5}
a
5
.
\newline
Answer:
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If
a
1
=
4
,
a
2
=
1
a_{1}=4, a_{2}=1
a
1
=
4
,
a
2
=
1
and
a
n
=
2
a
n
−
1
+
3
a
n
−
2
a_{n}=2 a_{n-1}+3 a_{n-2}
a
n
=
2
a
n
−
1
+
3
a
n
−
2
then find the value of
a
4
a_{4}
a
4
.
\newline
Answer:
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If
a
1
=
4
,
a
2
=
5
a_{1}=4, a_{2}=5
a
1
=
4
,
a
2
=
5
and
a
n
=
2
a
n
−
1
+
a
n
−
2
a_{n}=2 a_{n-1}+a_{n-2}
a
n
=
2
a
n
−
1
+
a
n
−
2
then find the value of
a
4
a_{4}
a
4
.
\newline
Answer:
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If
a
1
=
0
,
a
2
=
3
a_{1}=0, a_{2}=3
a
1
=
0
,
a
2
=
3
and
a
n
=
3
a
n
−
1
−
a
n
−
2
a_{n}=3 a_{n-1}-a_{n-2}
a
n
=
3
a
n
−
1
−
a
n
−
2
then find the value of
a
4
a_{4}
a
4
.
\newline
Answer:
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If
a
1
=
2
,
a
2
=
0
a_{1}=2, a_{2}=0
a
1
=
2
,
a
2
=
0
and
a
n
=
2
a
n
−
1
−
3
a
n
−
2
a_{n}=2 a_{n-1}-3 a_{n-2}
a
n
=
2
a
n
−
1
−
3
a
n
−
2
then find the value of
a
5
a_{5}
a
5
.
\newline
Answer:
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If
a
1
=
5
,
a
2
=
0
a_{1}=5, a_{2}=0
a
1
=
5
,
a
2
=
0
and
a
n
=
a
n
−
1
+
3
a
n
−
2
a_{n}=a_{n-1}+3 a_{n-2}
a
n
=
a
n
−
1
+
3
a
n
−
2
then find the value of
a
5
a_{5}
a
5
.
\newline
Answer:
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If
a
1
=
0
,
a
2
=
3
a_{1}=0, a_{2}=3
a
1
=
0
,
a
2
=
3
and
a
n
=
2
a
n
−
1
−
a
n
−
2
a_{n}=2 a_{n-1}-a_{n-2}
a
n
=
2
a
n
−
1
−
a
n
−
2
then find the value of
a
4
a_{4}
a
4
.
\newline
Answer:
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If
a
1
=
5
,
a
2
=
1
a_{1}=5, a_{2}=1
a
1
=
5
,
a
2
=
1
and
a
n
=
3
a
n
−
1
−
2
a
n
−
2
a_{n}=3 a_{n-1}-2 a_{n-2}
a
n
=
3
a
n
−
1
−
2
a
n
−
2
then find the value of
a
6
a_{6}
a
6
.
\newline
Answer:
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If
a
1
=
5
,
a
2
=
2
a_{1}=5, a_{2}=2
a
1
=
5
,
a
2
=
2
and
a
n
=
3
a
n
−
1
+
2
a
n
−
2
a_{n}=3 a_{n-1}+2 a_{n-2}
a
n
=
3
a
n
−
1
+
2
a
n
−
2
then find the value of
a
4
a_{4}
a
4
.
\newline
Answer:
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If
a
1
=
2
,
a
2
=
5
a_{1}=2, a_{2}=5
a
1
=
2
,
a
2
=
5
and
a
n
=
2
a
n
−
1
−
a
n
−
2
a_{n}=2 a_{n-1}-a_{n-2}
a
n
=
2
a
n
−
1
−
a
n
−
2
then find the value of
a
5
a_{5}
a
5
.
\newline
Answer:
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If
a
1
=
4
,
a
2
=
2
a_{1}=4, a_{2}=2
a
1
=
4
,
a
2
=
2
and
a
n
=
2
a
n
−
1
+
a
n
−
2
a_{n}=2 a_{n-1}+a_{n-2}
a
n
=
2
a
n
−
1
+
a
n
−
2
then find the value of
a
6
a_{6}
a
6
.
\newline
Answer:
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If
a
1
=
2
,
a
2
=
1
a_{1}=2, a_{2}=1
a
1
=
2
,
a
2
=
1
and
a
n
=
2
a
n
−
1
+
a
n
−
2
a_{n}=2 a_{n-1}+a_{n-2}
a
n
=
2
a
n
−
1
+
a
n
−
2
then find the value of
a
5
a_{5}
a
5
.
\newline
Answer:
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If
f
(
1
)
=
0
,
f
(
2
)
=
5
f(1)=0, f(2)=5
f
(
1
)
=
0
,
f
(
2
)
=
5
and
f
(
n
)
=
2
f
(
n
−
1
)
−
2
f
(
n
−
2
)
f(n)=2 f(n-1)-2 f(n-2)
f
(
n
)
=
2
f
(
n
−
1
)
−
2
f
(
n
−
2
)
then find the value of
f
(
5
)
f(5)
f
(
5
)
.
\newline
Answer:
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If
f
(
1
)
=
2
,
f
(
2
)
=
1
f(1)=2, f(2)=1
f
(
1
)
=
2
,
f
(
2
)
=
1
and
f
(
n
)
=
f
(
n
−
1
)
−
f
(
n
−
2
)
f(n)=f(n-1)-f(n-2)
f
(
n
)
=
f
(
n
−
1
)
−
f
(
n
−
2
)
then find the value of
f
(
5
)
f(5)
f
(
5
)
.
\newline
Answer:
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If
f
(
1
)
=
0
,
f
(
2
)
=
3
f(1)=0, f(2)=3
f
(
1
)
=
0
,
f
(
2
)
=
3
and
f
(
n
)
=
2
f
(
n
−
1
)
−
f
(
n
−
2
)
f(n)=2 f(n-1)-f(n-2)
f
(
n
)
=
2
f
(
n
−
1
)
−
f
(
n
−
2
)
then find the value of
f
(
6
)
f(6)
f
(
6
)
.
\newline
Answer:
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If
f
(
1
)
=
4
,
f
(
2
)
=
1
f(1)=4, f(2)=1
f
(
1
)
=
4
,
f
(
2
)
=
1
and
f
(
n
)
=
f
(
n
−
1
)
−
f
(
n
−
2
)
f(n)=f(n-1)-f(n-2)
f
(
n
)
=
f
(
n
−
1
)
−
f
(
n
−
2
)
then find the value of
f
(
5
)
f(5)
f
(
5
)
.
\newline
Answer:
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If
f
(
1
)
=
2
,
f
(
2
)
=
5
f(1)=2, f(2)=5
f
(
1
)
=
2
,
f
(
2
)
=
5
and
f
(
n
)
=
3
f
(
n
−
1
)
−
2
f
(
n
−
2
)
f(n)=3 f(n-1)-2 f(n-2)
f
(
n
)
=
3
f
(
n
−
1
)
−
2
f
(
n
−
2
)
then find the value of
f
(
5
)
f(5)
f
(
5
)
.
\newline
Answer:
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If
f
(
1
)
=
3
,
f
(
2
)
=
1
f(1)=3, f(2)=1
f
(
1
)
=
3
,
f
(
2
)
=
1
and
f
(
n
)
=
f
(
n
−
1
)
−
3
f
(
n
−
2
)
f(n)=f(n-1)-3 f(n-2)
f
(
n
)
=
f
(
n
−
1
)
−
3
f
(
n
−
2
)
then find the value of
f
(
6
)
f(6)
f
(
6
)
.
\newline
Answer:
Get tutor help
If
f
(
1
)
=
5
,
f
(
2
)
=
0
f(1)=5, f(2)=0
f
(
1
)
=
5
,
f
(
2
)
=
0
and
f
(
n
)
=
3
f
(
n
−
1
)
−
f
(
n
−
2
)
f(n)=3 f(n-1)-f(n-2)
f
(
n
)
=
3
f
(
n
−
1
)
−
f
(
n
−
2
)
then find the value of
f
(
6
)
f(6)
f
(
6
)
.
\newline
Answer:
Get tutor help
If
f
(
1
)
=
1
,
f
(
2
)
=
0
f(1)=1, f(2)=0
f
(
1
)
=
1
,
f
(
2
)
=
0
and
f
(
n
)
=
f
(
n
−
1
)
−
3
f
(
n
−
2
)
f(n)=f(n-1)-3 f(n-2)
f
(
n
)
=
f
(
n
−
1
)
−
3
f
(
n
−
2
)
then find the value of
f
(
6
)
f(6)
f
(
6
)
.
\newline
Answer:
Get tutor help
If
f
(
1
)
=
4
,
f
(
2
)
=
5
f(1)=4, f(2)=5
f
(
1
)
=
4
,
f
(
2
)
=
5
and
f
(
n
)
=
2
f
(
n
−
1
)
+
2
f
(
n
−
2
)
f(n)=2 f(n-1)+2 f(n-2)
f
(
n
)
=
2
f
(
n
−
1
)
+
2
f
(
n
−
2
)
then find the value of
f
(
5
)
f(5)
f
(
5
)
.
\newline
Answer:
Get tutor help
If
f
(
1
)
=
7
f(1)=7
f
(
1
)
=
7
and
f
(
n
+
1
)
=
3
f
(
n
)
−
5
f(n+1)=3 f(n)-5
f
(
n
+
1
)
=
3
f
(
n
)
−
5
then find the value of
f
(
5
)
f(5)
f
(
5
)
.
\newline
Answer:
Get tutor help
If
f
(
1
)
=
5
,
f
(
2
)
=
4
f(1)=5, f(2)=4
f
(
1
)
=
5
,
f
(
2
)
=
4
and
f
(
n
)
=
3
f
(
n
−
1
)
−
f
(
n
−
2
)
f(n)=3 f(n-1)-f(n-2)
f
(
n
)
=
3
f
(
n
−
1
)
−
f
(
n
−
2
)
then find the value of
f
(
5
)
f(5)
f
(
5
)
.
\newline
Answer:
Get tutor help
If
f
(
1
)
=
7
f(1)=7
f
(
1
)
=
7
and
f
(
n
+
1
)
=
−
3
f
(
n
)
−
4
f(n+1)=-3 f(n)-4
f
(
n
+
1
)
=
−
3
f
(
n
)
−
4
then find the value of
f
(
4
)
f(4)
f
(
4
)
\newline
Answer:
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