Learning Objectives
- Use the multiplication property of equality
- Solve algebraic equations using the multiplication property of equality
Solve algebraic equations using the multiplication property of equality
Just as you can add or subtract the same exact quantity on both sides of an equation, you can also multiply both sides of an equation by the same quantity to write an equivalent equation. Let’s look at a numeric equation, [latex]5\cdot3=15[/latex], to start. If you multiply both sides of this equation by 2, you will still have a true equation.
[latex]\begin{array}{r}5\cdot 3=15\,\,\,\,\,\,\, \\ 5\cdot3\cdot2=15\cdot2 \\ 30=30\,\,\,\,\,\,\,\end{array}[/latex]
This characteristic of equations is generalized in the multiplication property of equality.
Multiplication Property of Equality
For all real numbers a, b, and c: If a = b, then [latex]a\cdot{c}=b\cdot{c}[/latex] (or ab = ac).
If two expressions are equal to each other and you multiply both sides by the same number, the resulting expressions will also be equivalent.
When the equation involves multiplication or division, you can “undo” these operations by using the inverse operation to isolate the variable. When the operation is multiplication or division, your goal is to change the coefficient to 1, the multiplicative identity.
Example
Solve [latex]3x=24[/latex]. When you are done, check your solution.
You can also multiply the coefficient by the multiplicative inverse (reciprocal) in order to change the coefficient to 1.
Example
Solve [latex] \frac{1}{2 }{ x }={ 8}[/latex] for x.
In the video below you will see examples of how to use the multiplication property of equality to solve one-step equations with integers and fractions.
In the next example, we will solve a one-step equation using the multiplication property of equality. You will see that the variable is part of a fraction in the given equation, and using the multiplication property of equality allows us to remove the variable from the fraction. Remember that fractions imply division, so you can think of this as the variable k is being divided by 10. To “undo” the division, you can use multiplication to isolate k. Lastly, note that there is a negative term in the equation, so it will be important to think about the sign of each term as you work through the problem. Stop after each step you take to make sure all the terms have the correct sign.
Example
Solve [latex]-\frac{7}{2}=\frac{k}{10}[/latex] for k.
In the following video you will see examples of using the multiplication property of equality to solve a one-step equation involving negative fractions.
Summary
Equations are mathematical statements that combine two expressions of equal value. An algebraic equation can be solved by isolating the variable on one side of the equation using the properties of equality. To check the solution of an algebraic equation, substitute the value of the variable into the original equation.