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College Algebra/Solving Linear Equations and Inequalities/Properties of Equality

Solving Linear Equations and Inequalities

Contents

Properties of Equality

Addition Property of Equality

The addition property of equality can be used to undo subtraction when solving equations.
Linear equations are the simplest types of equations that contain variables. In a linear equation in one variable, each term can be written as a number or a product of the variable and a number. The variable is not raised to a power, in the denominator of a fraction, or under a radical, such as a square root. The variable can be multiplied or divided by a number, called a constant, or a number may be added to or subtracted from the variable.

Examples of Linear and Nonlinear Equations

Linear Equations Nonlinear Equations
x+3=7x+3=7
x2+4x=5x^2+4x=5
2x=42x=4
x+3=2\sqrt{x+3}=2
x5+1=23\frac{x}{5}+1=\frac{2}{3}
3x=x\frac{3}{x}=x
6x+1=4x−36x+1=4x-3
10x=10010^x=100

A solution of an equation is any value of the variable that makes the equation true, which means both sides are equal. Solving an equation means to find all possible solutions. The most common method of solving linear equations in one variable is to use properties of equality to isolate the variable, or rewrite the equation with the variable alone on one side.

The addition property of equality states that the solution of an equation does not change after adding the same number to both sides of the equation.

Addition Property of Equality Example
If the same quantity is added to both sides of a true equation, the resulting equation is still true. If a=ba=b, then:
a+c=b+ca+c=b+c

Step-By-Step Example
Applying the Addition Property of Equality
Solve the equation and then check whether the solution is true.
x−5=8x-5=8
Step 1
Use the addition property of equality. Add 5 to both sides of the equation to undo the subtraction.
x−5=8x−5+5=8+5\begin{aligned}x-5&=8\\x-5+5&=8+5 \end{aligned}
Step 2
Simplify by combining like terms.
x−5+5=8+5x=13\begin{aligned}x-5+5&=8+5\\x&=13 \end{aligned}
Step 3
Substitute xx for 13 to determine whether the solution is true.
x−5=813−5=88=8\begin{aligned}x-5&=8\\13-5&=8\\8&=8\end{aligned}
Solution
The equation is true. So, xx is equal to 13.

Subtraction Property of Equality

The subtraction property of equality can be used to undo addition when solving equations.
The subtraction property of equality states that the solution of an equation does not change after subtracting the same number from both sides of the equation.
Subtraction Property of Equality Example
If the same quantity is subtracted from both sides of a true equation, the resulting equation is still true. If a=ba=b, then:
a−c=b−ca-c=b-c

Step-By-Step Example
Applying the Subtraction Property of Equality
Solve the equation and then check whether the solution is true.
x+4=12x+4=12
Step 1
Use the subtraction property of equality. Subtract 4 from both sides to undo the addition:
x+4=12x+4−4=12−4\begin{aligned}x+4&=12\\x+4-4&=12-4\end{aligned}
Step 2
Simplify by combining like terms:
x+4−4=12−4x=8\begin{aligned}x+4-4&=12-4\\x&=8\end{aligned}
Step 3
Substitute xx for 8 in the original equation to determine whether the solution is true.
x+4=128+4=1212=12\begin{aligned}x+4&=12\\8+4&=12\\12&=12\end{aligned}
Solution
The equation is true. So, xx is equal to 8.

Multiplication Property of Equality

The multiplication property of equality can be used to undo division when solving equations.
The multiplication property of equality states that the solution of an equation does not change after multiplying both sides of an equation by the same nonzero number. Note that multiplying both sides of the equation by zero produces 0=00=0, which has no solutions.
Multiplication Property of Equality Example
If both sides of a true equation are multiplied by the same nonzero quantity, the resulting equation is still true. If a=ba=b and c≠0c\neq0, then:
ac=bcac=bc

Step-By-Step Example
Applying the Multiplication Property of Equality
Solve the equation and then check whether the solution is true.
x5=7\frac{x}{5}=7
Step 1
Use the multiplication property of equality. Multiply both sides by 5 to undo the division:
x5=7(x5)(5)=(7)(5)\begin{aligned}\frac{x}{5}&=7\\\left(\frac{x}{5}\right)(5)&=(7) (5)\end{aligned}
Step 2
Simplify the equation.
(x5)(5)=(7)(5)x=35\begin{aligned}\left(\frac{x}{5}\right)(5)&=(7) (5)\\x&=35\end{aligned}
Step 3
Substitute xx for 35 to determine whether the solution is true.
x5=7355=77=7\begin{aligned}\frac{x}{5}&=7\\\frac{35}{5}&=7\\7&=7 \end{aligned}
Solution
The equation is true. So, the value of xx is 35.

Division Property of Equality

The division property of equality can be used to undo multiplication when solving equations.
The division property of equality states that the solution of an equation does not change after dividing both sides of an equation by the same nonzero number. Division is the inverse operation of multiplication, which means that dividing a variable by a number will undo multiplication by the same number. So the division property of equality is used to solve equations in which a variable is multiplied by a number.
Division Property of Equality Example
If both sides of a true equation are divided by the same nonzero quantity, the resulting equation is still true. If a=ba=b and c≠0c\neq0, then:
ac=bc\frac{a}{c}=\frac{b}{c}

Step-By-Step Example
Applying the Division Property of Equality
Solve the equation and then check whether the solution is true.
2x=182x=18
Step 1
Use the division property of equality. Divide both sides by 2 to undo the multiplication:
2x=182x2=182\begin{aligned}2x&=18\\ \frac{2x}{2}&=\frac{18}{2}\end{aligned}
Step 2
Simplify the equation.
2x2=182x=9\begin{aligned} \frac{2x}{2}&=\frac{18}{2}\\x&=9\end{aligned}
Step 3
Substitute xx for 9 to determine whether the solution is true.
2x=18(2)(9)=1818=18\begin{aligned}2x&=18\\(2)(9)&=18\\18&=18 \end{aligned}
Solution
The equation is true. So, the value of xx is 9.
<Vocabulary>Solving Multistep Equations