Question

which system of equations has the same solution as the system below?
$4x + 2y = 58$
$x + 4y = 46$
answer
$\boldsymbol{-8x - 4y = 58}$
$x + 4y = 46$
$\boldsymbol{-8x + 2y = -116}$
$x + 4y = 46$
$\boldsymbol{4x - 4y = -116}$
$x + 4y = 46$
$\boldsymbol{-8x - 4y = -116}$
$x + 4y = 46$

Explanation:

Step1: Recall the property of systems of equations

A system of equations has the same solution if we perform a valid transformation (like multiplying an equation by a non - zero constant) on one of the equations. For the given system \(

$$\begin{cases}4x + 2y=58\\x + 4y = 46\end{cases}$$

\), we can multiply the first equation \(4x + 2y = 58\) by \(- 2\).

Step2: Multiply the first equation by -2

When we multiply the equation \(4x+2y = 58\) by \(-2\), we use the distributive property \(a(b + c)=ab+ac\). So, \(-2\times(4x + 2y)=-2\times58\).
Calculating the left - hand side: \(-2\times4x+(-2)\times2y=-8x - 4y\)
Calculating the right - hand side: \(-2\times58=-116\)
So the transformed first equation is \(-8x - 4y=-116\), and the second equation remains \(x + 4y = 46\). So the system \(

$$\begin{cases}-8x - 4y=-116\\x + 4y = 46\end{cases}$$

\) has the same solution as the original system.

Answer:

\(\boldsymbol{-8x - 4y=-116}\)
\(\boldsymbol{x + 4y = 46}\) (the fourth option in the given choices)