How can I solve equations like 2x + 3 = 8x + 21?

Questions like this ask you to solve an equation; this means to find a value of x that when put into the equation allows each side of the equation to be equal.

There are three important rules to follow when answering this type of question:

1. The operation you do to one side of the equation must be done to the other side (e.g. if you subtract 3 from the left hand side of the equals sign, you must subtract 3 from the right side, in order to ensure both sides of the equation remain equal.)

2. Group all like terms together (e.g. group numbers with other numbers, and group letters with letters (our unknown variable)).

3. Your aim is to get all of the letters (unknown variables) to one side of the equals sign, and all numbers on the other side of the equals sign, and then simplify.

Back to our question...

Solve 2x + 3 = 8x + 21

We need to consider how we can get all the letters on one side of the equals sign, and all numbers on the other side of the equals sign. If we subtract 2x from both sides (rule 1), then we are a step closer to accomplishing this.

2x + 3 = 8x + 21

• 2x        -2x

⇒ 3 = 8x -2x + 21

By rule 2, we can group the unknown variables together: 8x - 2x = 6x, so we can just write 6x for 8x - 2x.

⇒ 3 = 6x + 21

Now, we need to consider how we can get the 21 on the right hand side of the equation onto the left side of the equation. This means that we will have accomplished rule 3; having all the numbers on one side of the equals sign, and all the letters on the other! If we subtract 21 from the right hand side, we just get 6x on the right hand side, which accomplishes rule 3! Rule 1 says that we must subtract 21 from the left hand side too...

3 = 6x + 21

-21        -21

⇒ 3 - 21 = 6x

By rule 2, we can group the numbers on the left hand side together (3 - 21 = -18, so we can replace 3 - 21 by -18)

⇒ -18 = 6x

We now know what 6x is equal to, but we just want to find what x is equal to.

If we divide each side of the equation by 6, by rule 1, we just get x = , so we have solved our equation

-18 = 6x

÷6      ÷6

⇒ -3 = x

Our solution is x = -3.

We can check this by substituting x = -3, and seeing if the two sides of the equation are equal:

2x + 3 = 8x + 21

⇒ (2* -3) + 3 = (8* -3) + 21

⇒ -6 + 3 = -24 + 21

⇒ -3 = -3

Therefore, x = -3 is the correct solution to this equation.

Here are a few more questions to test your understanding, scroll down for the answers!

1. Solve 9x + 4 = 6x + 16

2. Solve -2 -13x = 7x + 18

3. Solve 4x + 2x - 6 = -3x + 12

Answers:

1. x = 4

2. x = -1

3. x = 2