We are now ready to “get to the good stuff.” You have the basics down and are ready to begin one of the most important topics in algebra: solving equations. The applications are limitless and extend to all careers and fields. Also, the skills and techniques you learn here will help improve your critical thinking and problemsolving skills. This is a great benefit of studying mathematics and will be useful in your life in ways you may not see right now.
Solve Equations Using the Subtraction and Addition Properties of Equality
We began our work solving equations in previous chapters. It has been a while since we have seen an equation, so we will review some of the key concepts before we go any further.
We said that solving an equation is like discovering the answer to a puzzle. The purpose in solving an equation is to find the value or values of the variable that make each side of the equation the same. Any value of the variable that makes the equation true is called a solution to the equation. It is the answer to the puzzle.
Definition: Solution of an Equation
A solution of an equation is a value of a variable that makes a true statement when substituted into the equation.
In the earlier sections, we listed the steps to determine if a value is a solution. We restate them here.
HOW TO: DETERMINE WHETHER A NUMBER IS A Solution TO AN EQUATION
Step 1. Substitute the number for the variable in the equation.
Step 2. Simplify the expressions on both sides of the equation.
Step 3. Determine whether the resulting equation is true.
 If it is true, the number is a solution.
 If it is not true, the number is not a solution.
Example \(\PageIndex{1}\):
Determine whether y = \(\dfrac{3}{4}\) is a solution for 4y + 3 = 8y.
Solution
Substitute \(\textcolor{red}{\dfrac{3}{4}}\) for y. 
$$4 \left(\textcolor{red}{\dfrac{3}{4}}\right) + 3 \stackrel{?}{=} 8 \left(\textcolor{red}{\dfrac{3}{4}}\right)$$ 
Multiply. 
$$3 + 3 \stackrel{?}{=} 6$$ 
Add. 
$$6 = 6\; \checkmark$$ 
Since y = \(\dfrac{3}{4}\) results in a true equation, \(\dfrac{3}{4}\) is a solution to the equation 4y + 3 = 8y.
Exercise \(\PageIndex{1}\):
Is y = \(\dfrac{2}{3}\) a solution for 9y + 2 = 6y?
 Answer

no
Exercise \(\PageIndex{2}\):
Is y = \(\dfrac{2}{5}\) a solution for 5y − 3 = 10y?
 Answer

no
We introduced the Subtraction and Addition Properties of Equality in Solving Equations Using the Subtraction and Addition Properties of Equality. In that section, we modeled how these properties work and then applied them to solving equations with whole numbers. We used these properties again each time we introduced a new system of numbers. Let’s review those properties here.
Definition: Subtraction and Addition Properties of Equality
Subtraction Property of Equality
For all real numbers a, b, and c, if a = b, then a − c = b − c.
Addition Property of Equality
For all real numbers a, b, and c, if a = b, then a + c = b + c.
When you add or subtract the same quantity from both sides of an equation, you still have equality.
We introduced the Subtraction Property of Equality earlier by modeling equations with envelopes and counters. Figure \(\PageIndex{1}\) models the equation x + 3 = 8.
Figure \(\PageIndex{1}\)
The goal is to isolate the variable on one side of the equation. So we ‘took away’ 3 from both sides of the equation and found the solution x = 5.
Some people picture a balance scale, as in Figure \(\PageIndex{2}\), when they solve equations.
Figure \(\PageIndex{2}\)
The quantities on both sides of the equal sign in an equation are equal, or balanced. Just as with the balance scale, whatever you do to one side of the equation you must also do to the other to keep it balanced.
Let’s review how to use Subtraction and Addition Properties of Equality to solve equations. We need to isolate the variable on one side of the equation. And we check our solutions by substituting the value into the equation to make sure we have a true statement.
Example \(\PageIndex{2}\):
Solve: x + 11 = −3.
Solution
To isolate x, we undo the addition of 11 by using the Subtraction Property of Equality.
Subtract 11 from each side to "undo" the addition. 
$$x + 11 \textcolor{red}{11} = 3 \textcolor{red}{11}$$ 
Simplify. 
$$x = 14$$ 
Check:
Substitute x = −14. 
$$\textcolor{red}{14} + 11 \stackrel{?}{=} 3$$ 

$$3 = 3\; \checkmark$$ 
Since x = −14 makes x + 11 = −3 a true statement, we know that it is a solution to the equation.
Exercise \(\PageIndex{3}\):
Solve: x + 9 = −7.
 Answer

x = 16
Exercise \(\PageIndex{4}\):
Solve: x + 16 = −4.
 Answer

x = 20
In the original equation in the previous example, 11 was added to the x, so we subtracted 11 to ‘undo’ the addition. In the next example, we will need to ‘undo’ subtraction by using the Addition Property of Equality.
Example \(\PageIndex{3}\):
Solve: m + 4 = −5.
Solution
Add 4 to each side to "undo" the subtraction. 
$$m + 4 \textcolor{red}{4} = 5 \textcolor{red}{4}$$ 
Simplify. 
$$m = 9$$ 
Check:
Substitute m = −9. 
$$\textcolor{red}{9} + 4 \stackrel{?}{=} 5$$ 

$$5 = 5\; \checkmark$$ 
The solution to m + 4 = −5 is m = −9.
Exercise \(\PageIndex{5}\):
Solve: n − 6 = −7.
 Answer

n = 1
Exercise \(\PageIndex{6}\):
Solve: x − 5 = −9.
 Answer

x = 4
Now let’s review solving equations with fractions.
Example \(\PageIndex{4}\):
Solve: n − \(\dfrac{3}{8}\) = \(\dfrac{1}{2}\).
Solution
Use the Addition Property of Equality. 
$$n  \dfrac{3}{8} \textcolor{red}{+ \dfrac{3}{8}} = \dfrac{1}{2} \textcolor{red}{+ \dfrac{3}{8}}$$ 
Find the LCD to add the fractions on the right. 
$$n  \dfrac{3}{8} + \dfrac{3}{8} = \dfrac{4}{8} + \dfrac{3}{8}$$ 
Simplify. 
$$n = \dfrac{7}{8}$$ 
Check:
Substitute n = \(\textcolor{red}{\dfrac{7}{8}}\). 
$$\textcolor{red}{\dfrac{7}{8}}  \dfrac{3}{8} \stackrel{?}{=} \dfrac{1}{2}$$ 
Subtract. 
$$\dfrac{4}{8} \stackrel{?}{=} \dfrac{1}{2}$$ 
Simplify. 
$$\dfrac{1}{2} = \dfrac{1}{2}\; \checkmark$$ 
The solution checks.
Exercise \(\PageIndex{7}\):
Solve: p − \(\dfrac{1}{3}\) = \(\dfrac{5}{6}\).
 Answer

\(p = \frac{7}{6}\)
Exercise \(\PageIndex{8}\):
Solve: q − \(\dfrac{1}{2}\) = \(\dfrac{1}{6}\).
 Answer

\( q = \frac{2}{3}\)
In Solve Equations with Decimals, we solved equations that contained decimals. We’ll review this next.
Example \(\PageIndex{5}\):
Solve a − 3.7 = 4.3.
Solution
Use the Addition Property of Equality. 
$$a  3.7 \textcolor{red}{+3.7} = 4.3 \textcolor{red}{+3.7}$$ 
Add. 
$$a = 8$$ 
Check:
Substitute a = 8. 
$$\textcolor{red}{8}  3.7 \stackrel{?}{=} 4.3$$ 
Simplify. 
$$4.3 = 4.3\; \checkmark$$ 
The solution checks.
Exercise \(\PageIndex{9}\):
Solve: b − 2.8 = 3.6.
 Answer

b = 6.4
Exercise \(\PageIndex{10}\):
Solve: c − 6.9 = 7.1.
 Answer

c = 14