## 0.5 Linear Equations with One Variable

### Quick reference

• For all real numbers a, b, and c: If $a=b$, then $a+c=b+c$.
• Multiplication Property of Equalities
• For all real numbers a, b, and c: If a = b, then $a\cdot{c}=b\cdot{c}$ (or ab = ac).
• Absolute Value
• For any positive number a, the solution of $\left|x\right|=a$ is

$x=a$ or $x=−a$

x can be a single variable or any algebraic expression.

• The Distributive Property of Multiplication
• For all real numbers a, b, and c, $a(b+c)=ab+ac$.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.

## Solve an algebraic equation using the addition property of equality

First, let’s define some important terminology:

• variables:  variables are symbols that stand for an unknown quantity, they are often represented with letters, like x, y, or z.
• coefficient: Sometimes a variable is multiplied by a number. This number is called the coefficient of the variable. For example, the coefficient of 3x is 3.
• term: a single number, or variables and numbers connected by multiplication. -4, 6x and $x^2$ are all terms
• expression: groups of terms connected by addition and subtraction.  $2x^2-5$ is an expression
• equation:  an equation is a mathematical statement that two expressions are equal. An equation will always contain an equal sign with an expression on each side. Think of an equal sign as meaning “the same as.” Some examples of equations are $y = mx +b$,  $\frac{3}{4}r = v^{3} - r$, and  $2(6-d) + f(3 +k) = \frac{1}{4}d$

The following figure shows how coefficients, variables, terms, and expressions all come together to make equations. In the equation $2x-3^2=10x$, the variable is $x$, a coefficient is $10$, a term is $10x$, an expression is $2x-3^2$.

Equation made of coefficients, variables, terms and expressions.

### Using the Addition Property of Equality

An important property of equations is that you can add the same quantity to both sides of an equation and still maintain an equivalent equation.

For all real numbers a, b, and c: If $a=b$, then $a+c=b+c$.

If two expressions are equal to each other, and you add the same value to both sides of the equation, the equation will remain equal.

### Solve algebraic equations using the addition property of equality

When you solve an equation, you find the value of the variable that makes the equation true. In order to solve the equation, you isolate the variable. Isolating the variable means rewriting an equivalent equation in which the variable is on one side of the equation and everything else is on the other side of the equation.

### Examples

Solve $x-6=8$.

Solve $x+5=27$.

Since subtraction can be written as addition (adding the opposite), the addition property of equality can be used for subtraction as well. So just as you can add the same value to each side of an equation without changing the meaning of the equation, you can subtract the same value from each side of an equation.

### Examples

Solve $x+10=-65$. Check your solution.

Solve $x-4=-32$. Check your solution.

It is always a good idea to check your answer whether you are requested to or not.

The examples above are sometimes called one-step equations because they require only one step to solve. In these examples, you either added or subtracted a constant from both sides of the equation to isolate the variable and solve the equation.

With any equation, you can check your solution by substituting the value for the variable in the original equation. In other words, you evaluate the original equation using your solution. If you get a true statement, then your solution is correct.

## Solving One-Step Equations Containing Absolute Values with Addition

The absolute value of a number or expression describes its distance from 0 on a number line. Since the absolute value expresses only the distance, not the direction of the number on a number line, it is always expressed as a positive number or 0.

For example, $−4$ and 4 both have an absolute value of 4 because they are each 4 units from 0 on a number line—though they are located in opposite directions from 0 on the number line.

When solving absolute value equations and inequalities, you have to consider both the behavior of absolute value and the properties of equality and inequality.

Because both positive and negative values have a positive absolute value, solving absolute value equations means finding the solution for both the positive and the negative values.

Let’s first look at a very basic example.

$\displaystyle \left| x \right|=5$

This equation is read “the absolute value of x is equal to five.” The solution is the value(s) that are five units away from 0 on a number line.

You might think of 5 right away; that is one solution to the equation. Notice that $−5$ is also a solution because $−5$ is 5 units away from 0 in the opposite direction. So, the solution to this equation $\displaystyle \left| x \right|=5$ is $x = −5$ or $x = 5$.

### Solving Equations of the Form $|x|=a$

For any positive number a, the solution of $\left|x\right|=a$ is

$x=a$ or $x=−a$

x can be a single variable or any algebraic expression.

You can solve a more complex absolute value problem in a similar fashion.

### Example

Solve $\displaystyle \left| x+5\right|=15$.

## 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, $5\cdot3=15$, to start. If you multiply both sides of this equation by 2, you will still have a true equation.

$\begin{array}{r}5\cdot 3=15\,\,\,\,\,\,\, \\ 5\cdot3\cdot2=15\cdot2 \\ 30=30\,\,\,\,\,\,\,\end{array}$

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 $a\cdot{c}=b\cdot{c}$ (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 $3x=24$. 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 $\frac{1}{2 }{ x }={ 8}$ for x.

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 $-\frac{7}{2}=\frac{k}{10}$ for k.

## Solving One-Step Equations Containing Absolute Values With Multiplication

Remember that absolute value refers to the distance from zero. You can use the same technique of first isolating the absolute value, then setting up and solving two equations to solve an absolute value equation involving multiplication.

### Example

Solve $\displaystyle \left| 2x\right|=6$.

### Example

Solve $\displaystyle\frac{1}{3}\left|k\right|=12$.

## Use properties of equality to isolate variables and solve algebraic equations

Steps With an End In Sight

There are some equations that you can solve in your head quickly. For example—what is the value of y in the equation $2y=6$? Chances are you didn’t need to get out a pencil and paper to calculate that $y=3$. You only needed to do one thing to get the answer: divide 6 by 2.

Other equations are more complicated. Solving $\displaystyle 4\left( \frac{1}{3}t+\frac{1}{2}\right)=6$ without writing anything down is difficult! That’s because this equation contains not just a variable but also fractions and terms inside parentheses. This is a multi-step equation, one that takes several steps to solve. Although multi-step equations take more time and more operations, they can still be simplified and solved by applying basic algebraic rules.

You can think of an equation as a balance scale, with the goal being to rewrite the equation so that it is easier to solve but still balanced. The addition property of equality and the multiplication property of equality explain how you can keep the scale, or the equation, balanced. Whenever you perform an operation to one side of the equation, if you perform the same exact operation to the other side, you’ll keep both sides of the equation equal.

If the equation is in the form $ax+b=c$, where x is the variable, you can solve the equation as before. First “undo” the addition and subtraction, and then “undo” the multiplication and division.

### Example

Solve $3y+2=11$.

### Example

Solve $3x+5x+4-x+7=88$.

Some equations may have the variable on both sides of the equal sign, as in this equation: $4x-6=2x+10$.

To solve this equation, we need to “move” one of the variable terms. This can make it difficult to decide which side to work with. It doesn’t matter which term gets moved, $4x$ or $2x$, however, to avoid negative coefficients, you can move the smaller term.

### Examples

Solve: $4x-6=2x+10$

## Solving Multi-Step Equations With Absolute Value

We can apply the same techniques we used for solving a one-step equation which contains absolute value to an equation that will take more than one step to solve.  Let’s start with an example where the first step is to write two equations, one equal to positive 26 and one equal to negative 26.

### Example

Solve for p. $\left|2p–4\right|=26$

Now let’s look at an example where you need to do an algebraic step or two before you can write your two equations. The goal here is to get the absolute value on one side of the equation by itself. Then we can proceed as we did in the previous example.

### Example

Solve for w. $3\left|4w–1\right|–5=10$

## 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 to use 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, $a(b+c)=ab+ac$.

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 $a$. $4\left(2a+3\right)=28$

In the next example, you will see that there are parentheses on both sides of the equal sign, so you will need to use the distributive property twice. Notice that you are going to need to distribute a negative number, so be careful with negative signs!

### Example

Solve for $t$.  $2\left(4t-5\right)=-3\left(2t+1\right)$

Sometimes, you will encounter a multi-step equation with fractions. If you prefer not working with fractions, you can use the multiplication property of equality to multiply both sides of the equation by a common denominator of all of the fractions in the equation. This will clear all the fractions out of the equation. See the example below.

### Example

Solve  $\frac{1}{2}x-3=2-\frac{3}{4}x$ by clearing the fractions in the equation first.

Of course, if you like to work with fractions, you can just apply your knowledge of operations with fractions and solve.

Regardless of which method you use to solve equations containing variables, you will get the same answer. You can choose the method you find the easiest! Remember to check your answer by substituting your solution into the original equation.

Sometimes, you will encounter a multi-step equation with decimals. If you prefer not working with decimals, you can use the multiplication property of equality to multiply both sides of the equation by a factor of 10 that will help clear the decimals. See the example below.

### Example

Solve $3y+10.5=6.5+2.5y$ by clearing the decimals in the equation first.

Here are some steps to follow when you solve multi-step equations.

### 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.