In the system of equations below, c is a constant. \begin{aligned} 3x - 2y &= 3c - 1 \\ x … — SAT

In the system of equations below, $c$ is a constant.

\begin{aligned} 3x - 2y &= 3c - 1 \\ x + 3y &= 2c - 8 \end{aligned}

If the solution $(x, y)$ to the system satisfies $x + y = 2$ , what is the value of $c$ ?

4Answer
B
16
C
3
D
-1

Answer

The correct value of the constant

c

is 4.

To find the value of

c

, the linear constraint

x + y = 2

is rearranged to

x = 2 - y

. Substituting this expression into the first equation yields

3(2 - y) - 2y = 3c - 1

, which simplifies to

6 - 5y = 3c - 1

, or

y = \frac{7 - 3c}{5}

. Substituting it into the second equation yields

(2 - y) + 3y = 2c - 8

, which simplifies to

2 + 2y = 2c - 8

, or

y = c - 5

. Equating these two expressions gives

c - 5 = \frac{7 - 3c}{5}

. Multiplying both sides by

5

results in

5c - 25 = 7 - 3c

. Adding

3c

and

25

to both sides yields

8c = 32

, which gives

c = 4

Step-by-Step Solution

Express

x

in terms of

y

using the given constraint

x + y = 2

x = 2 - y

This allows us to substitute the relation into the system of equations and reduce the variables from two to one.

Substitute

x = 2 - y

into the first equation,

3x - 2y = 3c - 1

, and isolate

y

3(2 - y) - 2y = 3c - 1 \Rightarrow 6 - 5y = 3c - 1 \Rightarrow 5y = 7 - 3c \Rightarrow y = \frac{7 - 3c}{5}

To represent

y

as a function of the parameter

c

Substitute

x = 2 - y

into the second equation,

x + 3y = 2c - 8

, and isolate

y

(2 - y) + 3y = 2c - 8 \Rightarrow 2 + 2y = 2c - 8 \Rightarrow 2y = 2c - 10 \Rightarrow y = c - 5

To obtain another independent expression for

y

in terms of

c

Equate the two expressions for

y

and solve for

c

c - 5 = \frac{7 - 3c}{5} \Rightarrow 5(c - 5) = 7 - 3c \Rightarrow 5c - 25 = 7 - 3c \Rightarrow 8c = 32 \Rightarrow c = 4

Since both expressions represent the same value

y

, they must be equal, allowing us to determine the constant

c

Key Concept

Solving a system of linear equations containing unknown parameters under given linear constraints.

Alternative Method

Alternatively, substitute the relationship

x = 2 - y

into both equations to write them as a system in terms of

y

and

c

3(2 - y) - 2y = 3c - 1 \Rightarrow 5y + 3c = 7

and

(2 - y) + 3y = 2c - 8 \Rightarrow 2y - 2c = -10

. This simplified system can be solved for

c

by multiplying the first equation by 2 and the second equation by 5 to eliminate

y

2(5y + 3c) - 5(2y - 2c) = 2(7) - 5(-10) \Rightarrow 6c + 10c = 14 + 50 \Rightarrow 16c = 64 \Rightarrow c = 4

Estimated Time:2m 0s

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

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