Solving Linear Equations and Inequalities

Solving Multistep Equations

Two-Step Equations

Properties of equality can be combined to solve two-step equations.

In some linear equations, more than one operation has been applied to the variable. In this case, the properties of equality can be combined to isolate the variable. To isolate the variable, undo the operations applied to the variable in reverse order:

1. Apply the addition or subtraction property of equality to undo subtraction or addition.

2. Apply the multiplication or division property of equality to undo division or multiplication.

Step-By-Step Example
Solving Two-Step Equations
Solve the equation:
x3+5=2\frac{x}{3}+5=-2
Step 1
Apply the subtraction property of equality to undo the addition. Subtract 5 from both sides to isolate the variable.
x3+5=2x3+55=25x3=7\begin{aligned}\frac{x}{3}+5&=-2\\\frac{x}{3}+5-5&=-2-5\\\frac{x}{3}&=-7 \end{aligned}
Step 2
Apply the multiplication property of equality to undo the division. Multiply both sides by 3 to isolate the variable.
3x3=3(7)x=21\begin{aligned}3\cdot\frac{x}{3}&=3\cdot(-7)\\x&=-21\end{aligned}
Solution
The solution is x=21x=-21.

Multistep Equations

Properties of equality and properties of operations can be combined to solve equations using multiple steps.

Some equations require more than two steps to solve. Properties of operations may be used to simplify one or both sides of the equation before applying the properties of equality to isolate the variable.

1. Simplify the expressions on each side of the equation.

2. Isolate the variable term using the addition or subtraction property of equality.

3. Isolate the variable using the multiplication or division property of equality.

Simplifying the expressions on each side of an equation may involve applying the distributive property. The distributive property states that multiplying an expression by a sum is the same as multiplying the expression by each term in the sum and then adding the products:
a(b+c)=ab+aca(b+c)=ab+ac
Step-By-Step Example
Solving Multistep Equations
Solve the equation:
11=3(x4)+211=3(x-4)+2
Step 1
Simplify the right side of the equation by applying the distributive property. Multiply 3 by xx and –4.
11=3(x4)+211=3x12+211=3x10\begin{aligned}11&=3(x-4)+2\\11&=3x-12+2\\11&=3x-10\end{aligned}
Step 2
Apply the addition property of equality to undo the subtraction. Add 10 to both sides.
11=3x1011+10=3x10+1021=3x\begin{aligned}11&=3x-10\\11+10&=3x-10+10\\21&=3x\end{aligned}
Step 3
Apply the division property of equality to undo the multiplication. Divide both sides by 3.
21=3x213=3x37=x\begin{aligned}21&=3x\\ \frac{21}{3}&=\frac{3x}{3}\\ 7&=x\end{aligned}
Solution
The solution is x=7x=7.

Equations with Variables on Both Sides

Properties of equality can be used to solve equations where the variable appears on both sides.
To solve equations with variables on both sides, start by simplifying each side, if necessary. Apply the addition or subtraction property of equality so the variable appears on one side of the equation only. Then solve the equation using properties of equality.
Step-By-Step Example
Solving Equations with Variables on Both Sides
Solve the equation:
5y+8=3y125y+8=3y-12
Step 1
The variable terms are 5y5y and 3y3y. Apply the subtraction property of equality to combine variable terms on one side of the equation. Subtract 3y3y from both sides.
5y+8=3y125y+83y=3y123y2y+8=12\begin{aligned}5y+8&=3y-12\\5y+8-3y&=3y-12-3y\\2y+8&=-12\end{aligned}
Step 2
Apply the subtraction property of equality to undo the addition. Subtract 8 from both sides.
2y+8=122y+88=1282y=20\begin{aligned}2y+8&=-12\\2y+8-8&=-12-8\\2y&=-20\end{aligned}
Solution
Apply the division property of equality to undo the multiplication. Divide both sides by 2.
2y=202y2=202y=10\begin{aligned}2y&=-20\\\frac{2y}{2}&=\frac{-20}{2}\\y&=-10\end{aligned}