Solve Equations with the Distributive Property
Learning Outcomes
- Use the properties of equality and the distributive property to solve equations containing parentheses
- Clear fractions and decimals from equations to make them easier to solve
The Distributive Property
As we solve linear equations, we often need to do some work to write the linear equations in a form we are familiar with solving. This section will focus on manipulating an equation we are asked to solve in such a way that we can use the skills we learned for solving multi-step equations to ultimately arrive at the solution. Parentheses can make solving a problem difficult, if not impossible. To get rid of these unwanted parentheses we have the distributive property. Using this property we multiply the number in front of the parentheses by each term inside of the parentheses.The Distributive Property of Multiplication
For all real numbers a, b, and c, [latex]a(b+c)=ab+ac[/latex]. What this means is that when a number multiplies an expression inside parentheses, you can distribute the multiplication to each term of the expression individually. Then, you can follow the steps we have already practiced to isolate the variable and solve the equation.Example
Solve for [latex]a[/latex]. [latex]4\left(2a+3\right)=28[/latex]Answer: Apply the distributive property to expand [latex]4\left(2a+3\right)[/latex] to [latex]8a+12[/latex]
[latex]\begin{array}{r}4\left(2a+3\right)=28\\ 8a+12=28\end{array}[/latex]
Subtract [latex]12[/latex] from both sides to isolate the variable term.[latex]\begin{array}{r}8a+12\,\,\,=\,\,\,28\\ \underline{-12\,\,\,\,\,\,-12}\\ 8a\,\,\,=\,\,\,16\end{array}[/latex]
Divide both terms by [latex]8[/latex] to get a coefficient of [latex]1[/latex].[latex]\begin{array}{r}\underline{8a}=\underline{16}\\8\,\,\,\,\,\,\,\,\,\,\,\,8\,\,\\a\,=\,\,2\end{array}[/latex]
Example
Solve for [latex]t[/latex]. [latex]2\left(4t-5\right)=-3\left(2t+1\right)[/latex]Answer: Apply the distributive property to expand [latex]2\left(4t-5\right)[/latex] to [latex]8t-10[/latex] and [latex]-3\left(2t+1\right)[/latex] to[latex]-6t-3[/latex]. Be careful in this step—you are distributing a negative number, so keep track of the sign of each number after you multiply.
[latex]\begin{array}{r}2\left(4t-5\right)=-3\left(2t+1\right)\,\,\,\,\,\, \\ 8t-10=-6t-3\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\end{array}[/latex]
Add [latex]6t[/latex] to both sides to begin combining like terms.[latex]\begin{array}{r}8t-10=-6t-3\\ \underline{+6t\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,+6t}\,\,\,\,\,\,\,\\ 14t-10=\,\,\,\,-3\,\,\,\,\,\,\,\end{array}[/latex]
Add 10 to both sides of the equation to isolate t.[latex]\begin{array}{r}14t-10=-3\\ \underline{+10\,\,\,+10}\\ 14t=\,\,\,7\,\end{array}[/latex]
The last step is to divide both sides by 14 to completely isolate t.[latex]\begin{array}{r}\dfrac{14t}{14}\normalsize=\dfrac{7}{14}\end{array}[/latex]
We simplify the fraction [latex]\dfrac{7}{14}[/latex] into the final answer of [latex]t=\dfrac{1}{2}[/latex]
Example
Solve [latex]\dfrac{1}{2}\normalsize x-3=2-\dfrac{3}{4}\normalsize x[/latex] by clearing the fractions in the equation first.Answer: Multiply both sides of the equation by [latex]4[/latex], the common denominator of the fractional coefficients.
[latex]\begin{array}{r}\dfrac{1}{2}\normalsize{x-3=2-}\dfrac{3}{4}\normalsize{x}\,\,\,\,\,\,\,\,\,\,\,\,\,\\\\ 4\left(\dfrac{1}{2}\normalsize{x-3}\right)=4\left(2-\dfrac{3}{4}\normalsize{x}\right)\end{array}[/latex]
Use the distributive property to expand the expressions on both sides. Multiply.[latex]\begin{array}{r}4\left(\dfrac{1}{2}\normalsize{x}\right)-4\left(3\right)=4\left(2\right)-4\left(-\dfrac{3}{4}\normalsize{x}\right)\\\\ \dfrac{4}{2}\normalsize{x}-12=8-\dfrac{12}{4}\normalsize{x}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \\\\ 2x-12=8-3x\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \end{array}[/latex]
Add 3x to both sides to move the variable terms to only one side.[latex]\begin{array}{r}2x-12=8-3x\, \\\underline{+3x\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,+3x}\\ 5x-12=8\,\,\,\,\,\,\,\,\,\,\,\,\,\end{array}[/latex]
Add 12 to both sides to move the constant terms to the other side.[latex]\begin{array}{r}5x-12=8\,\,\\ \underline{\,\,\,\,\,\,+12\,+12} \\5x=20\end{array}[/latex]
Divide to isolate the variable.[latex]\begin{array}{r}\underline{5x}=\underline{20}\\ 5\,\,\,\,\,\,\,\,\,5\,\,\\ x=4\end{array}[/latex]
Example
Solve [latex]3y+10.5=6.5+2.5y[/latex] by clearing the decimals in the equation first.Answer: Since the smallest decimal place value represented in the equation is 0.10, we want to multiply by 10 to clear the decimals from the equation.
[latex]\begin{array}{r}3y+10.5=6.5+2.5y\,\,\,\,\,\,\,\,\,\,\,\,\\\\ 10\left(3y+10.5\right)=10\left(6.5+2.5y\right)\end{array}[/latex]
Use the distributive property to expand the expressions on both sides.[latex]\begin{array}{r}10\left(3y\right)+10\left(10.5\right)=10\left(6.5\right)+10\left(2.5y\right)\end{array}[/latex]
Multiply.[latex]30y+105=65+25y[/latex]
Move the smaller variable term, [latex]25y[/latex], by subtracting it from both sides.[latex]\begin{array}{r}30y+105=65+25y\,\,\\ \underline{-25y\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,-25y} \\5y+105=65\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\end{array}[/latex]
Subtract 105 from both sides to isolate the term with the variable.[latex]\begin{array}{r}5y+105=65\,\,\,\\ \underline{\,\,\,\,\,\,-105\,-105} \\5y=-40\end{array}[/latex]
Divide both sides by 5 to isolate the y.[latex]\begin{array}{r}\underline{5y}=\underline{-40}\\ 5\,\,\,\,\,\,\,\,\,\,\,\,\,5\,\,\\ y=-8\,\,\end{array}[/latex]
Solving Multi-Step Equations
1. (Optional) Multiply to clear any fractions or decimals. 2. Simplify each side by clearing parentheses and combining like terms. 3. Add or subtract to isolate the variable term—you may have to move a term with the variable. 4. Multiply or divide to isolate the variable. 5. Check the solution.Summary
Complex, multi-step equations often require multi-step solutions. Before you can begin to isolate a variable, you may 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. If your multi-step equation has an absolute value, you will need to solve two equations, sometimes isolating the absolute value expression first.Licenses & Attributions
CC licensed content, Original
- Revision and Adaptation. Provided by: Lumen Learning License: CC BY: Attribution.
- Solving an Equation with One Set of Parentheses. Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. License: CC BY: Attribution.
- Solving an Equation with Parentheses on Both Sides. Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. License: CC BY: Attribution.
- Solving an Equation with Fractions (Clear Fractions). Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. License: CC BY: Attribution.
- Solving an Equation with Decimals (Clear Decimals). Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. License: CC BY: Attribution.
CC licensed content, Shared previously
- Unit 10: Solving Equations and Inequalities, from Developmental Math: An Open Program. Provided by: Monterey Institute of Technology and Education Located at: https://www.nroc.org/. License: CC BY: Attribution.