In multi-step equations, you will need to make use of the techniques used in solving one-step
equations. You may want to review those topics before beginning the examples in this lesson.
Just as with solving one-step or two-step or any equation, one goal in solving an equation
is to have only variables on one side
of the equal sign and numbers on the other side
of the equal sign. The other goal is to have the number in front of the variable
equal to one. Keep in mind that the variable
does not always have to be x. These equations can make use of any letter as a variable.
The strategy for getting the variable
by itself with a coefficient
of 1 involves using opposite operations. For example, to move something that is added to the other side
of the equation, you should subtract. The most important thing to remember in solving a linear equation
is that whatever you do to one side
of the equation, you MUST do to the other side. So if you subtract a number from one side, you MUST subtract the same value from the other side. You will see how this works in the examples.
This problem is typical of what you can expect to see in a multi-step equation. This equation has a variable on both sides of the equal sign. We must first put the variables on the same side. Let’s move the 2x from the right side to the left side by subtracting 2x from both sides.
Now that we have isolated the variable on the left side of the equation, we can go about solving the new equation using techniques of solving one and two step equations.
We must first make sure we have the variable only on one side. It does not matter which side we choose. Some people prefer to always move the variable to the left side while others try to make sure they don’t have a negative coefficient. There is no “always correct” method. The most important thing is to know how to get the variable only on one side.
Before we begin to use any of our equation solving skills, we must first simplify the equation by using the distributive property and eliminating the parentheses.
Now we simplify by combining like terms.
At this point, we can use our equation solving skills to find the solution to the equation.
Be very careful with the last example. It is very easy to make an error with the signs either when working with the -3z terms or when dividing by -6. Be careful not to make arithmetic mistakes that can cause an error with the sign of your answer.