Solving Multi-step Equations
· Simplify algebraic equations using the Properties of Equality and the Distributive Property to clear parentheses and fractions.
There are some equations that you can solve in your head, in an instant. If I asked you to solve 2y = 4, chances are you wouldn’t need to get out a pencil and paper to calculate that y = 2. You only needed to do one thing to get the answer, divide by 2.
Other equations are more complicated. Can you solve without writing anything down? Me neither. That’s because it contains not just a variable but also fractions and terms inside parentheses. This is a multi-step equation, one that takes several operations to solve. Don’t worry though. Although they take more time and more operations, multi-step equations can still be simplified and solved by applying basic algebraic rules. Two helpful tools are the Distributive Property and multiplication by a common denominator.
Just as with simpler equations, solving multi-step equations usually means isolating a variable on one side of the equals sign. Step-by-step, we must get the variable out of parentheses, away from other terms, and with a coefficient of 1. If a variable is inside parentheses, they can be cleared by applying the Distributive Property of Multiplication.
The Distributive Property states that for all real numbers a, b, and c, . What that means is that when a number multiplies an expression inside parentheses, we can distribute the multiplication to each term of the expression individually. Let’s go back to the equation . We can apply the distributive property and clear the parentheses by multiplying each term inside of them by 6. The expression then becomes . Now there are no parentheses.
We’ve seen how to simplify a multi-step equation by applying the Distributive Property to clear parentheses. It’s also helpful to simplify equations that include fractions before we try to solve them. We can do this by multiplying both sides by a common denominator of all of the fractions in the equation.
Let’s go back to the equation . The lowest common denominator of the fractions and is 8. So we’ll multiply the expression with those fractions by 8. Because we must preserve the equality of the equation, we’ll multiply the expression on the other side by 8, too. Now we’ve got .
But wait—we just put parentheses back into the equation again. Now what? Simple, we go back to our old friend the Distributive Property once again. Once we multiply each term inside the parentheses by 8, both the parentheses and the fractions will disappear. Sweet.
Complex, multi-step equations often require multi-step solutions. Before we can begin to isolate a variable, we might need to simplify the equation first. This may mean using the Distributive Property to remove parentheses, or multiplying both sides of an equation by a common denominator to get rid of fractions. Sometimes it requires both techniques.