Question

which of the following is a solution to the system made up by ( x + y = 27 ) and ( y = 2x )?
(1) ( x = 2 ) and ( y = 20 )
(3) ( x = 5 ) and ( y = 10 )
(2) ( x = 10 ) and ( y = 17 )
(4) ( x = 9 ) and ( y = 18 )
is ( (-2, 5) ) a solution to the system shown below? justify your answer.

Explanation:

Step1: Substitute \( y = 2x \) into \( x + y = 27 \)

We know the system of equations is \(

$$\begin{cases}x + y = 27\\y = 2x\end{cases}$$

\). Substitute \( y = 2x \) into the first equation, we get \( x + 2x=27 \).

Step2: Solve for \( x \)

Combine like terms: \( 3x = 27 \). Then divide both sides by 3: \( x=\frac{27}{3}=9 \).

Step3: Solve for \( y \)

Substitute \( x = 9 \) into \( y = 2x \), we get \( y = 2\times9 = 18 \).

Step4: Check each option

• Option (1): \( x = 2,y = 20 \). Check \( y = 2x \): \( 2\times2 = 4

eq20 \), so not a solution.

• Option (2): \( x = 10,y = 17 \). Check \( y = 2x \): \( 2\times10 = 20

eq17 \), so not a solution.

• Option (3): \( x = 5,y = 10 \). Check \( x + y \): \( 5 + 10 = 15

eq27 \), so not a solution.

• Option (4): \( x = 9,y = 18 \). Check \( x + y \): \( 9+18 = 27 \), and \( y = 2x \): \( 2\times9 = 18 \), so it satisfies both equations.

Answer:

(4) \( x = 9 \) and \( y = 18 \)