Consider the system of equations below. \begin{aligned} 3(x + y) - 2(x - y) &= 20 \\ 2(x + y) + 3(x - y) &= 22 \end{aligned} If (x, y) is the solution to the system of equations, what is the value of x?

Consider the system of equations below. \begin{aligned} 3(x + y) - 2(x - y) &= 20 \\ 2(x +… — SAT

Consider the system of equations below.

\begin{aligned} 3(x + y) - 2(x - y) &= 20 \\ 2(x + y) + 3(x - y) &= 22 \end{aligned}

(x, y)

is the solution to the system of equations, what is the value of

x

Answer: 5

Answer

The value of

x

5

Expanding and simplifying the given system yields

x + 5y = 20

and

5x - y = 22

. Multiplying the second equation by 5 results in

25x - 5y = 110

. Adding this to the first equation eliminates

y

, leaving

26x = 130

. Dividing both sides by 26 gives

x = 5

Step-by-Step Solution

Expand the algebraic expressions in both equations to rewrite the system in standard form.

x + 5y = 20

and

5x - y = 22

Expanding the first equation gives

3x + 3y - 2x + 2y = 20

, which simplifies to

x + 5y = 20

. Expanding the second equation gives

2x + 2y + 3x - 3y = 22

, which simplifies to

5x - y = 22

Multiply the second equation by 5 to align the coefficients of the y-terms.

25x - 5y = 110

Multiplying

5x - y = 22

by 5 allows the subtraction of

y

to cancel with the addition of

5y

in the first equation.

Add the first equation to the modified second equation to eliminate the y-variable and solve for x.

26x = 130

, which simplifies to

x = 5

Adding

(x + 5y) + (25x - 5y) = 20 + 110

yields

26x = 130

. Dividing by 26 gives the final value of

x

Key Concept

Solving systems of linear equations using expansion and elimination.

Alternative Method

Define substitution variables

u = x + y

and

v = x - y

. The system simplifies to

3u - 2v = 20

and

2u + 3v = 22

. Multiplying the first equation by 3 and the second by 2 gives

9u - 6v = 60

and

4u + 6v = 44

. Adding these equations yields

13u = 104 \implies u = 8

. Substituting

u = 8

back in gives

16 + 3v = 22 \implies 3v = 6 \implies v = 2

. Now, solve the system

x + y = 8

and

x - y = 2

. Adding these two equations gives

2x = 10 \implies x = 5

Estimated Time:1m 30s

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

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