Question

The point (2, 1) is a solution to which of the following systems of equations?

2x + y = 5 and −2x + y = 2
−2x − y = −5 and 2x − y = 2
−x + y = 3 and −3x + y = −5
x + y = 3 and 3x − y = 5

Answer 1

`\huge\boxed{\left \{ {{x+y=3} \atop {3x-y=5}} \right.}`

We can solve this problem by testing the solution against each system of equations. We'll do this by substituting the point into each equation for each system.

System 1 -
`\Large\textbf{X}`

`\begin{array}c\textbf{First Equation}&\textbf{Second Equation}\\\cline{1-2}\begin{aligned}2x+y&=5\\2(2)+1&=5\\4+1&=5\\5&=5\\&\checkmark\end{aligned}&\begin{aligned}-2x+y&=2\\-2(2)+1&=2\\-4+1&=2\\-3&=2\\&\text{X}\end{aligned}\end{array}`

System 2 -
`\Large\textbf{X}`

`\begin{array}c\textbf{First Equation}&\textbf{Second Equation}\\\cline{1-2}\begin{aligned}-2x-y&=-5\\-2(2)-1&=-5\\-4-1&=-5\\-5&=-5\\&\checkmark\end{aligned}&\begin{aligned}2x-y&=2\\2(2)-1&=2\\4-1&=2\\3&=2\\&\text{X}\end{aligned}\end{array}`

System 3 -
`\Large\textbf{X}`

`\begin{array}c\textbf{First Equation}&\textbf{Second Equation}\\\cline{1-2}\begin{aligned}-x+y&=3\\-2+1&=3\\-1&=3\\&\text{X}\end{aligned}&\begin{aligned}-3x+y&=-5\\-3(2)+1&=-5\\-6+1&=-5\\-5&=-5\\&\checkmark\end{aligned}\end{array}`

System 4 -
`\Large\checkmark`

`\begin{array}c\textbf{First Equation}&\textbf{Second Equation}\\\cline{1-2}\begin{aligned}x+y&=3\\2+1&=3\\3&=3\\&\checkmark\end{aligned}&\begin{aligned}3x-y&=5\\3(2)-1&=5\\6-1&=5\\5&=5\\&\checkmark\end{aligned}\end{array}`

Since both equations in the final system are true when solved with
`(2, 1)`, the answer is the last system.