Lesson 2.5: Equations

New Concept

An equation states that two expressions are equal. We'll find a 'solution'—the value of a variable that makes the equation true. You'll master solving equations by isolating the variable using opposite operations, like addition and subtraction.

What’s next

Next, you'll apply this with interactive examples and practice cards to master solving one-step equations.

Property

An equation is a mathematical statement that two algebraic expressions are equal. We can use variables to write equations. For example, $D = F + 6$ where $D$ and $F$ stand for Delbert’s age and Francine’s age. Finding relationships between variables and describing them as equations is an important algebraic skill.

Examples

• A store offers a 20 dollar discount on all jackets. An equation for the sale price, $S$, in terms of the original price, $P$, is $S = P - 20$.

• A company gives a 10% bonus on top of a base salary. An equation for the total pay, $T$, in terms of the base salary, $B$, is $T = 1.10B$.

Property

A solution of an equation is a value of the variable that makes the equation true. An equation containing a variable can be true or false, depending on the value substituted for the variable.

Examples

• To check if 15 is a solution to $8x = 120$, we substitute 15 for $x$. This gives $8(15) = 120$. Since $120 = 120$ is true, 15 is a solution.

• To check if 5 is a solution to $\frac{m}{4} = 2$, we substitute 5 for $m$. This gives $\frac{5}{4} = 2$. Since $1.25 = 2$ is false, 5 is not a solution.

Property

Each of the four arithmetic operations has an opposite operation that 'undoes' its effect. Addition and subtraction are opposite operations. Multiplication and division are opposite operations. For any numbers $x$, $a$, and $b$ (where $b \neq 0$):

Property

To solve an equation, we isolate the variable. The process is:

1. Ask yourself: Which operation has been performed on the variable?
2. Perform the opposite operation on both sides of the equation to maintain balance.
3. Check your solution by substituting it into the original equation.

Examples

• To solve $x - 5 = 12$, we undo the subtraction by adding 5 to both sides: $(x - 5) + 5 = 12 + 5$, which simplifies to $x = 17$.

• To solve $y + 9 = 20$, we undo the addition by subtracting 9 from both sides: $(y + 9) - 9 = 20 - 9$, which simplifies to $y = 11$.