show and explain how replacing one equation by the sum of that equation and a multiple of the other produces…

Explanation:

Step1: Multiply the second - equation

Multiply the second equation $4x - 14y=-68$ by 2. We get $2(4x - 14y)=2\times(-68)$, which simplifies to $8x-28y = - 136$.

Step2: Add the new equation to the first equation

Add the new equation $8x-28y=-136$ to the first equation $8x + 7y=39$. $(8x + 7y)+(8x-28y)=39+( - 136)$ $8x+7y + 8x-28y=39 - 136$ $16x-21y=-97$. The original system is $\begin{cases}8x + 7y=39\4x-14y=-68\end{cases}$, and the new system is $\begin{cases}8x + 7y=39\16x-21y=-97\end{cases}$. The reason the solutions are the same is that when we perform the operation of adding a multiple of one equation to another, we are essentially performing a valid algebraic transformation. If $(x_0,y_0)$ is a solution of the original system, it satisfies both $8x_0 + 7y_0=39$ and $4x_0-14y_0=-68$. When we create the new equation by adding a multiple of one equation to the other, $(x_0,y_0)$ will also satisfy the new - formed equation because of the properties of equality.

Answer:

By multiplying the second equation by 2 and adding it to the first equation, we get a new system with the same solutions as the original system. The new system is $\begin{cases}8x + 7y=39\16x-21y=-97\end{cases}$ and the operation is valid due to the properties of equality.