   SEARCH HOME Math Central Quandaries & Queries  Question from kristina, a student: hi im kristina and i cannot figure out how 2 do these multi-step equations 5x + 9 = 3x + 7 i can on getting the wrong answer according 2 the book Hi Kristina,

You want to start with the equation 5x + 9 = 3x + 7 and manipulate it to end with an equation of the form x = some number. At every stage of the manipulation you need to ensure that the two sides of the equation remain equal. If you add the same amount to each side the equation will remain valid and likewise if you subtract the same amount to each side the equation will remain valid. You can also multiply each side by the same amount or divide each side by the same amount as long as you don't divide by zero. Division by zero is not a valid operation. I am going to illustrate with

7x + 4 = 4x - 5.

I don't want x on the right side of the equation so the first operation I would perform is to subtract 4x from each side.

7x + 4 = 4x - 5
7x + 4 - 4x = 4x - 5 - 4x
3x + 4 = -5

It's looking better. On the right side is a number but on the left side I have 3x rather than x and there is the number 4 on the left side which I don't want. I could divide both sides by 3 to change the 3x to x but that would introduce fractions and I don't want fractions unless they are necessary. What I will do is subtract 4 from each side to obtain

3x + 4 = -5
3x + 4 - 4 = -5 - 4
3x = -9

Now I divide each side by 3 and get

3x = -9
3x/3 = -9/3
x = -3

I can check that my answer is correct by substituting x = -3 into each side of the original equation.

Left side: 7x + 4 = 7 × -3 + 4 = -21 + 4 = -17
Right side: 4x - 5 = 4 × -3 - 5 = -12 - 5 = -17

Hence x = -3 does satisfy the equation.

Now try 5x + 9 = 3x + 7.

Penny     Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.