What if you had an equation like
We’ve seen that when we solve for an unknown variable, it can take just one or two steps to get the terms in the right places. Now we’ll look at solving equations that take several steps to isolate the unknown variable. Such equations are referred to as multi-step equations.
In this section, we’ll simply be combining the steps we already know how to do. Our goal is to end up with all the constants on one side of the equation and all the variables on the other side. We’ll do this by collecting like terms. Don’t forget, like terms have the same combination of variables in them.
Before we can combine the variable terms, we need to get rid of that fraction.
First let’s put all the terms on the left over a common denominator of three:
Combining the fractions then gives us
Combining like terms in the numerator gives us
Multiplying both sides by 3 gives us
Subtracting 4 from both sides gives us
And finally, dividing both sides by -12 gives us
Solving Multi-Step Equations Using the Distributive Property
You may have noticed that when one side of the equation is multiplied by a constant term, we can either distribute it or just divide it out. If we can divide it out without getting awkward fractions as a result, then that’s usually the better choice, because it gives us smaller numbers to work with. But if dividing would result in messy fractions, then it’s usually better to distribute the constant and go from there.
The first thing we want to do here is get rid of the parentheses. We could use the Distributive Property, but it just so happens that 7 divides evenly into 21. That suggests that dividing both sides by 7 is the easiest way to solve this problem.
If we do that, we get
Once again, we want to get rid of those parentheses. We could divide both sides by 17, but that would give us an inconvenient fraction on the right-hand side. In this case, distributing is the easier way to go.
Distributing the 17 gives us
Before we can collect like terms, we need to get rid of the parentheses using the Distributive Property. That gives us
Next we add 37 to both sides to get
And finally, we divide both sides by -2 to get
Watch this video for help with the Examples above.
- If dividing a number outside of parentheses will produce fractions, it is often better to use the Distributive Property to expand the terms and then combine like terms to solve the equation.
Solve the following equation for
This function contains both fractions and decimals. We should convert all terms to one or the other. It’s often easier to convert decimals to fractions, but in this equation the fractions are easy to convert to decimals—and with decimals we don’t need to find a common denominator!
In decimal form, our equation becomes
Distributing to get rid of the parentheses, we get
Collecting and combining like terms gives us
Then we can subtract 1.82 from both sides to get
Solve the following equations for the unknown variable.