SAT

Given the equations below, what is the value of a?
7(2a + 3b) = 56
b + 2a = –4

–5
–4
–2
4
6

Solution

Solve the second equation for b in terms of a; b + 2a = –4, b = –2a – 4. Substitute this expression for b in the first equation and solve for a:

7(2a + 3(–2a – 4)) = 56

7(2a + –6a – 12) = 56

7(–4a – 12) = 56

–28a – 84 = 56

–28a = 140

a = –5

Given the equations below, what is one possible value of p?
4pq – 6 = 10
4p – 2q = –14

–2
¹⁄₂
1
2
8

Solution

Solve the second equation for q in terms of p; 4p – 2q = –14, –2q = –4p – 14, q = 2p + 7. Substitute this expression for q in the first equation and solve for p:

4pq – 6 = 10

4p(2p + 7) – 6 = 10

8p² + 28p – 6 = 10

8p² + 28p – 16 = 0

2p² + 7p – 4 = 0

(2p – 1)(p + 4) = 0

2p – 1 = 0, 2p = 1, p = ¹⁄₂

p + 4 = 0, p = –4

Given the equations below, what is the value of b?
9a – 2(b + 4) = 30
4¹⁄₂(a) – 3b = 3

2
4
6
8
10

Solution

In the first equation, multiply the (b + 4) term by –2: –2(b + 4) = –2b – 8. Add 8 to both sides of the equation, and the first equation becomes 9a – 2b = 38. Multiply the second equation by 2 and subtract it from the first equation. The a
terms will drop out, and you can solve for b:

2(4.5a – 3b = 3) = 9a – 6b = 6

b = 8

Given the equations below, what is the value of n?
2(m + n) + m = 9
3m – 3n = 24

–5
–3
–3
5
8

Solution

In the first equation, multiply the (m + n) term by 2 and add m: 2(m + n) + m = 2m + 2n + m = 3m + 2n. Subtract the second equation from the first equation. The m terms will drop out, and you can solve for n:

n = –3

Given the equations below, what is the value of y?
3x + 7y = 19
\(\frac{4y}{x}\) = 1

¹⁄₄
¹⁄₂
1
2
4

Solution

Solve the second equation for x in terms of y;\(\frac{4y}{x}\)= 1, x = 4y. Substitute this expression for x in the first equation and solve for y:

3x + 7y = 19

3(4y) + 7y = 19

12y + 7y = 19

19y = 19

y = 1

Given the equations below, what is the value of b?
–7a + ^b⁄₄ = 25
b + a = 13

–3
4
12
13
16

Solution

Solve the second equation for a in terms of b; b + a = 13, a = 13 – b. Substitute this expression for a in the first equation and solve for b:

–7a + ^b⁄₄ = 25

–7(13 – b) + ^b⁄₄ = 25

7b – 91 + ^b⁄₄ = 25

\(\frac{29b}{4}\) = 116

29b = 464

b = 16

Given the equations below, what is the value of x?
3(x + 4) – 2y = 5
2y – 4x = 8

–2
–1
1
13
15

Solution

In the first equation, multiply the (x + 4) term by 3: 3(x + 4) = 3x + 12. Then, subtract 12 from both sides of the equation, and the first equation becomes 3x – 2y = –7. Add the two equations together. The y terms will drop out, and you can solve for a:

x = –1

Given the equations below, what is one possible value of y?
xy = 32
2x – y = 0

–8
–2
2
4
16

Solution

Solve the first equation for x in terms of y; xy = 32, x =\(\frac{32}{y}\). Substitute this expression for x in the second equation and solve for y:

2x – y = 0

2(\(\frac{32}{y}\)) – y = 0

\(\frac{32}{y}\) – y = 0

\(\frac{64}{y}\) = y

y² = 64

y = –8, y = 8

Given the equations below, what is the value of b?
5a + 3b = –2
5a – 3b = –38

–6
–4
6
12
13

Solution

Add the two equations together. The b terms will drop out, and you can solve for a:

a = –4

Substitute –4 for a in the first equation and solve for b:

5(–4) + 3b = –2

–20 + 3b = –2

3b = 18

b = 6

Given the equations below, what is the value of x?
2x + y = 6
^y⁄₂ + 4x = 12

–2
Zero
1
3
6

Solution

Solve the first equation for y in terms of x; 2x + y = 6, y = 6 – 2x. Substitute this expression for y in the second equation and solve for x:

\(\frac{6-2x}{2+4x}=12\)

3 – x + 4x = 12

3x + 3 = 12

3x = 9

x = 3

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