Question

2. By substitution 2y + 2x =-4y + x =-5

Answer 1

We have to solve the following system by substitution:

`\mleft\{\begin{aligned}2y+2x=-4 \\ y+x=-5\end{aligned}\mright.`

When we have to solve a system via substitution, we clear out one variable on one equation and we replace it onto the second equation. In this example, we will clear out the variable y on the second equation (but you can try it with x also), then:

`\begin{gathered} y+x=-5 \\ y=-5-x \end{gathered}`

This answer will be substituted onto the first equation:

`2(-5-x)+2x=-4`

And we clear the variable x. We use the distributive property and the rules of an equality, we sum to both sides of the equation a value as shown:

`\begin{gathered} 2(-5-x)+2x=-4 \\ -10-2x+2x=-4 \\ -10+0=-4 \\ -10=-4 \end{gathered}`

This last equality is false. And as such, it means that the system above doesn't have solutions, the lines they represent are parallel and doesn't meet.

Note: We can also note that if you clear out the variable y, and treated as linear functions, the slope of both equations will be the same, which justifies why those two equations are parallel.