# Video: Finding the Solution Set of Linear Equations with an Unknown on Both Sides from a Given Substitution Set

Find the solution set of 2𝑥 + 2 = 𝑥 − 5 using the substitution set {−12, 7, −7, 3}.

03:30

### Video Transcript

Find the solution set of two 𝑥 plus two equals 𝑥 minus five using the substitution set minus 12, seven, minus seven, and three.

In order to solve this question, we need to substitute each of the numbers: minus 12, seven, minus seven, and three one at a time into the equation. Let’s start by substituting negative 12 into both sides of the equation. The left-hand side reads two multiplied by negative 12 add two. The right-hand side reads negative 12 take away five. Two multiplied by negative 12 gives us negative 24. When we add two to this, we get negative 22. On the right-hand side, negative 12 take away five gives us negative 17. As these numbers are not equal to each other, negative 12 is not a solution to the equation.

We’ll now substitute seven into both sides of the equation. On the left-hand side, we have two multiplied by seven add two, and on the right-hand side, we have seven subtract five. Two multiplied by seven is 14; add two gives us 16 on the left-hand side. Seven subtract five gives us an answer of two. So once again seven is not a solution to this equation.

The third number we need to substitute is negative seven. This gives us two multiplied by negative seven plus two. And on the right-hand side, negative seven take away five. Two multiplied by negative seven is negative 14. When we add two to this, we end up with negative 12. On the right-hand side, negative seven take away five also gives us negative 12. This means that negative seven is a solution to the equation two 𝑥 plus two equals 𝑥 minus five.

Finally, we are going to substitute three into our equation. On the left-hand side, this gives us two multiplied by three add two. And on the right-hand side, three subtract five. Two multiplied by three is six add two equals eight, whereas three take away five gives us negative two. As eight is not equal to negative two, then three is not a solution to this equation.

When we substituted in the four numbers into the equation, three of them give different answers on the left- and the right-hand side of the equal sign. Negative seven, however, gave us negative 12 on the left-hand side and negative 12 on the right-hand side. This means that the only solution to this equation from the substitution set negative 12, seven, negative seven, and three is negative seven.