Solving Multi-Step Equations

Learning Outcomes

Solve a linear equation that requires multiple steps and a combination of the properties of equality

Each of the first few sections of this chapter has dealt with solving one specific form of a linear equation. It’s time now to lay out an overall strategy that can be used to solve any linear equation. We call this the general strategy. Some equations won’t require all the steps to solve, but many will. Simplifying each side of the equation as much as possible first makes the rest of the steps easier.

general strategy for solving linear equations

Simplify each side of the equation as much as possible. Use the Distributive Property to remove any parentheses. Combine like terms.
If there are fractions or decimals in the equation, multiply by the least common denominator to clear them.
Collect all the variable terms to one side of the equation. Use the Addition or Subtraction Property of Equality.
Collect all the constant terms to the other side of the equation. Use the Addition or Subtraction Property of Equality.
Make the coefficient of the variable term equal to [latex]1[/latex]. Use the Multiplication or Division Property of Equality. State the solution to the equation.
Check the solution. Substitute the solution into the original equation, to make sure the result is a true statement.

Example

Solve: [latex]3\left(x+2\right)=18[/latex]

Solution:

	[latex]3(x+2)=18[/latex]
Simplify each side of the equation as much as possible. Use the Distributive Property.	[latex]3x+6=18[/latex]
Collect all variable terms on one side of the equation—all [latex]x[/latex] s are already on the left side.
Collect constant terms on the other side of the equation. Subtract [latex]6[/latex] from each side.	[latex]3x+6\color{red}{-6}=18\color{red}{-6}[/latex]
Simplify.	[latex]3x=12[/latex]
Make the coefficient of the variable term equal to [latex]1[/latex]. Divide each side by [latex]3[/latex].	[latex]\Large\frac{3x}{\color{red}{3}}\normalsize =\Large\frac{12}{\color{red}{3}}[/latex]
Simplify.	[latex]x=4[/latex]
Check:	[latex]3(x+2)=18[/latex]
Let [latex]x=4[/latex].	[latex]3(\color{red}{4}+2)\stackrel{\text{?}}{=}18[/latex]
	[latex]3(6)\stackrel{\text{?}}{=}18[/latex]
	[latex]18=18\quad\checkmark[/latex]

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Example

Solve: [latex]-\left(x+5\right)=7[/latex]

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Example

Solve: [latex]4\left(x - 2\right)+5=-3[/latex]

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Example

Solve: [latex]8 - 2\left(3y+5\right)=0[/latex]

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example

Solve: [latex]3\left(x - 2\right)-5=4\left(2x+1\right)+5[/latex]

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Example

Solve: [latex]\Large\frac{1}{2}\normalsize\left(6x - 2\right)=5-x[/latex]

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Watch the following video to see another example of how to solve an equation that requires distributing a fraction.

In the next video, we show an example of solving an equation that requires distributing a fraction. In this case, you will need to clear fractions after you distribute.

In many applications, we will have to solve equations with decimals. The same general strategy will work for these equations.

example

Solve: [latex]0.45\left(a+0.8\right)=0.3\left(a+2.2\right)[/latex]

Show Solution

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The following video provides another example of how to solve an equation that requires distributing a decimal.

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	[latex]-(x+5)=7[/latex]
Simplify each side of the equation as much as possible by distributing. The only [latex]x[/latex] term is on the left side, so all variable terms are on the left side of the equation.	[latex]-x-5=7[/latex]
Add [latex]5[/latex] to both sides to get all constant terms on the right side of the equation.	[latex]-x-5\color{red}{+5}=7\color{red}{+5}[/latex]
Simplify.	[latex]-x=12[/latex]
Make the coefficient of the variable term equal to [latex]1[/latex] by multiplying both sides by [latex]-1[/latex].	[latex]\color{red}{-1}(-x)=\color{red}{-1}(12)[/latex]
Simplify.	[latex]x=-12[/latex]
Check:	[latex]-(x+5)=7[/latex]
Let [latex]x=-12[/latex].	[latex]-(\color{red}{-12}+5)\stackrel{\text{?}}{=}7[/latex]
	[latex]-(-7)\stackrel{\text{?}}{=}7[/latex]
	[latex]7=7\quad\checkmark[/latex]

	[latex]4(x-2)+5=-3[/latex]
Simplify each side of the equation as much as possible. Distribute.	[latex]4x-8+5=-3[/latex]
Combine like terms	[latex]4x-3=-3[/latex]
The only [latex]x[/latex] is on the left side, so all variable terms are on one side of the equation.
Add [latex]3[/latex] to both sides to get all constant terms on the other side of the equation.	[latex]4x-3\color{red}{+3}=-3\color{red}{+3}[/latex]
Simplify.	[latex]4x=0[/latex]
Make the coefficient of the variable term equal to [latex]1[/latex] by dividing both sides by [latex]4[/latex].	[latex]\Large\frac{4x}{\color{red}{4}}\normalsize =\Large\frac{0}{\color{red}{4}}[/latex]
Simplify.	[latex]x=0[/latex]
Check:	[latex]4(x-2)+5=-3[/latex]
Let [latex]x=0[/latex].	[latex]4(\color{red}{0-2})+5\stackrel{\text{?}}{=}-3[/latex]
	[latex]4(-2)+5\stackrel{\text{?}}{=}-3[/latex]
	[latex]-8+5\stackrel{\text{?}}{=}-3[/latex]
	[latex]-3=-3\quad\checkmark[/latex]

	[latex]8-2(3y+5)=0[/latex]
Simplify—use the Distributive Property.	[latex]8-6y-10=0[/latex]
Combine like terms.	[latex]-6y-2=0[/latex]
Add [latex]2[/latex] to both sides to collect constants on the right.	[latex]-6y-2\color{red}{+2}=0\color{red}{+2}[/latex]
Simplify.	[latex]-6y=2[/latex]
Divide both sides by [latex]-6[/latex].	[latex]\Large\frac{-6y}{\color{red}{-6}}\normalsize =\Large\frac{2}{\color{red}{-6}}[/latex]
Simplify.	[latex]y=-\Large\frac{1}{3}[/latex]
Check:	[latex]8-2(3y+5)=0[/latex]
Let [latex]y=-\frac{1}{3}[/latex]	[latex]8-2(3(\color{red}{-\Large\frac{1}{3}}\normalsize )+5)\stackrel{\text{?}}{=}0[/latex]
	[latex]8-2(-1+5)\stackrel{\text{?}}{=}0[/latex]
	[latex]8-2(4)\stackrel{\text{?}}{=}0[/latex]
	[latex]8-8\stackrel{\text{?}}{=}0[/latex]
	[latex]0=0\quad\checkmark[/latex]

	[latex]3(x-2)-5=4(2x+1)+5[/latex]
Distribute.	[latex]3x-6-5=8x+4+5[/latex]
Combine like terms.	[latex]3x-11=8x+9[/latex]
Subtract [latex]3x[/latex] to get all the variables on the right, since [latex]8>3[/latex] .	[latex]3x\color{red}{-3x}-11=8x\color{red}{-3x}+9[/latex]
Simplify.	[latex]-11=5x+9[/latex]
Subtract [latex]9[/latex] to get the constants on the left.	[latex]-11\color{red}{-9}=5x+9\color{red}{-9}[/latex]
Simplify.	[latex]-20=5x[/latex]
Divide by [latex]5[/latex].	[latex]\Large\frac{-20}{\color{red}{5}}\normalsize =\Large\frac{5x}{\color{red}{5}}[/latex]
Simplify.	[latex]-4=x[/latex]
Check: Substitute: [latex]-4=x[/latex] .	[latex]3(\color{red}{-4}-2)-5\overset{?}{=}4(2(\color{red}{-4})+1)+5[/latex]
	[latex]3(-6)-5\overset{?}{=}4(-8+1)+5[/latex]
	[latex]-18-5\overset{?}{=}4(-7)+5[/latex]
	[latex]-23\overset{?}{=}-28+5[/latex]
	[latex]-23\overset{?}{=}-23\quad\checkmark[/latex]

	[latex]\Large\frac{1}{2}\normalsize(6x-2)=5-x[/latex]
Distribute.	[latex]3x-1=5-x[/latex]
Add [latex]x[/latex] to get all the variables on the left.	[latex]3x-1\color{red}{+x}=5-x\color{red}{+x}[/latex]
Simplify.	[latex]4x-1=5[/latex]
Add [latex]1[/latex] to get constants on the right.	[latex]4x-1\color{red}{+1}=5\color{red}{+1}[/latex]
Simplify.	[latex]4x=6[/latex]
Divide by [latex]4[/latex].	[latex]\Large\frac{4x}{\color{red}{4}}\normalsize =\Large\frac{6}{\color{red}{4}}[/latex]
Simplify.	[latex]x=\Large\frac{3}{2}[/latex]
Check: Let [latex]x=\Large\frac{3}{2}[/latex] .	[latex]\Large\frac{1}{2}\normalsize (6(\Large\frac{\color{red}{3}}{\color{red}{2}}\normalsize )-2)\overset{?}{=}5-(\Large\frac{\color{red}3}{\color{red}2})[/latex]
	[latex]\Large\frac{1}{2}\normalsize(9-2)\overset{?}{=}\Large\frac{10}{2}\normalsize -\Large\frac{3}{2}[/latex]
	[latex]\Large\frac{1}{2}\normalsize(7)\overset{?}{=}\Large\frac{7}{2}[/latex]
	[latex]\Large\frac{7}{2}\normalsize =\Large\frac{7}{2}\normalsize\quad\checkmark[/latex]

	[latex]0.45\left(a+0.8\right)=0.3\left(a+2.2\right)[/latex]
Distribute.	[latex]0.45a+0.36=0.3a+0.66[/latex]
Multiply by the least common denominator, 100	[latex]45a+36=30a+66[/latex]
Subtract [latex]30a[/latex] to get all the [latex]x[/latex] s to the left.	[latex]45a\color{red}{-30a}+36=30a+66\color{red}{-30a}[/latex]
Simplify.	[latex]15a+36=66[/latex]
Subtract [latex]36[/latex] to get the constants to the right.	[latex]15a+36\color{red}{-36}=66\color{red}{-36}[/latex]
Simplify.	[latex]15a=30[/latex]
Divide.	[latex]\large{\frac{15a}{15}=\frac{30}{15}}[/latex]
Simplify.	[latex]x = 2[/latex]
Check: Let [latex]x=2[/latex]	[latex]0.45\left(2+0.8\right)=0.3\left(2+2.2\right)[/latex]
	[latex]1.26=1.26\quad\checkmark[/latex]

Module 10: Linear Equations

Learning Outcomes

general strategy for solving linear equations

Example

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Example

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Example

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