Given the system of equations below: \begin{aligned} 3x - y &= 11 \\ 2x + 3y &= 11 \end{al… — SAT

Given the system of equations below:

\begin{aligned} 3x - y &= 11 \\ 2x + 3y &= 11 \end{aligned}

If $(x, y)$ is the solution to the system of equations above, what is the value of $x + 2y$ ?

A
$4$
$6$ Answer
C
$8$
D
$9$

Answer

The value of

x + 2y

6

To solve the system, we can isolate

y

in the first equation to get

y = 3x - 11

. Substituting this into the second equation gives

2x + 3(3x - 11) = 11

, which simplifies to

11x - 33 = 11

, or

11x = 44

, resulting in

x = 4

. Substituting

x = 4

back into the equation for

y

gives

y = 3(4) - 11 = 1

. The value of the expression

x + 2y

is therefore

4 + 2(1) = 6

Step-by-Step Solution

Isolate

y

in the first equation.

y = 3x - 11

To prepare for substitution into the second equation.

Substitute the expression for

y

into the second equation and solve for

x

2x + 3(3x - 11) = 11 \implies 11x - 33 = 11 \implies 11x = 44 \implies x = 4

To find the value of

x

Substitute the value of

x

back into the isolated equation for

y

to solve for

y

y = 3(4) - 11 \implies y = 1

To find the value of

y

Evaluate the expression

x + 2y

using the values of

x

and

y

4 + 2(1) = 6

To find the final requested value.

Key Concept

Solving a system of linear equations using substitution or elimination, and evaluating a linear combination of the variables.

Alternative Method

Alternatively, the system can be solved using the elimination method. Multiply the first equation by

3

to align the

y

-coefficients:

9x - 3y = 33

. Add this equation to the second equation,

2x + 3y = 11

, to eliminate

y

, yielding

11x = 44

, so

x = 4

. Then substitute

x = 4

into either original equation to find

y = 1

Estimated Time:1m 30s

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

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