|Solving Multi-Step Linear Equations
In this lesson, we are going to look at a few worked examples while putting emphasis on the key steps in solving multi-step equations. You will have the opportunity to practice on your own by trying some problems and compare your answers to the solutions provided. If you just want to practice and skip the lesson itself, go ahead, and click the button below.
To solve multi-step equations, you will still need the techniques you learned in solving one-step and two-step equations. This type of equation requires additional steps in order to solve for the value of the unknown variable. Usually the variable involved is x, but it is not always the case. It could be any letters such as m, n, h and z.
The main goal in solving multi-step equations is to keep the unknown variable on one side of the equal symbol while keeping the constant or pure number on the opposite side. More importantly, there is no rule where to keep the variable. It all depends on your preference. The "standard" way is to have it on the left side, but there are cases when it is convenient to leave it on the right side of the equation.
Finally, since we are dealing with equations, we need to keep in mind that whatever we do on one side must be applied to the other side to keep everything balanced. For instance, adding 5 on the left should force you to add 5 on the right side. To get rid of numbers in the process of solving equations, ALWAYS remember the idea of opposite operations because they are used to cancel or move around numbers.
Key steps to remember
1) Eliminate parenthesis by applying the Distributive Property
2) Simplify both sides of the equation by Combining Like Terms. In other words, combine similar variables and constants together.
3) Decide where you want to keep the variable; that helps you decide where to keep the constants (opposite side where the variable is located).
4) Cancel out numbers by applying opposite operations: addition and subtraction are opposite operations as in the case of multiplication and division.
Now it's time to take a look at some examples!
Example 1: Solve the multi-step equation
This is a typical problem in multi-step equations where there are variables on both sides. Notice that there are no parenthesis in this equation and nothing to combine like terms in either both sides of the equation. Clearly, our first step is to decide where to keep or isolate the unknown variable x. Since 7x is "larger" than 2x, then we might as well keep it on the left side.
This means we have to get rid of the 2x on the right side. To do that, we need to subtract both sides by 2x because the opposite of +2x is -2x.
After simplifying by subtracting both sides by 2x, we have...
It's nice to see just the variable x on the left side. This implies that we have to move all the constants to the right side and that +3 on the left must be removed. The opposite of +3 is -3, therefore, we will subtract both sides by 3.
After subtracting both sides by 3, we get...
The last step is to isolate variable x by itself on the left side of the equation. Since +5 is multiplying x, then its opposite operation is to divide by +5. So, we are going to divide both sides by 5 and then we are done!