Investigation 7: Balanced Equations

New Concept

Algebra is the language for finding unknown values. This course introduces the fundamental rule: whatever you do to one side of an equation, you must do to the other.

What’s next

Next, we'll use a balance scale model to visualize how to solve for unknown values using addition, subtraction, multiplication, and division.

Property

Equations are sometimes called balanced equations because the two sides of the equation “balance” each other. A balance scale can be used as a model of an equation, where the equal sign is the pivot point. For example, $x + 12 = 33$ is a balanced equation.

Examples

• The equation $x + 15 = 40$ is like a scale with $x + 15$ on one side and $40$ on the other.
• To find $x$, you must take away $15$ from both sides: $x + 15 - 15 = 40 - 15$.
• The scale stays perfectly balanced, showing the solution: $x = 25$.

Explanation

Think of an equation like a perfectly balanced scale. If you add or remove something from one side, you must do the exact same thing to the other side to keep it from tipping over! This single rule is the secret to solving any equation and finding the value of your unknown variable, like $x$.

Property

To solve an equation, you must isolate the variable. This means getting the variable (like $x$) all by itself on one side of the equal sign. To do this, you perform the inverse, or opposite, operation.

Examples

• To solve $x + 18 = 45$, undo the addition by subtracting $18$ from both sides: $x = 27$.
• To solve $2x = 132$, undo the multiplication by dividing both sides by $2$: $x = 66$.
• To solve $y - 7 = 10$, undo the subtraction by adding $7$ to both sides: $y = 17$.

Explanation

To get the variable alone, you have to undo whatever is happening to it. If a number is being added to your variable, you subtract it from both sides. If your variable is being multiplied by a number, you do the opposite and divide both sides by that number. It’s like a puzzle where you reverse the steps.

Property

The number multiplying the variable is called the coefficient.

Examples

• In the equation $4w = 132$, the coefficient of $w$ is $4$.
• For the term $1.2m$, the coefficient of $m$ is $1.2$.
• In the equation $\frac{3}{4}x = \frac{9}{10}$, the coefficient of $x$ is the fraction $\frac{3}{4}$.

Explanation

Think of the coefficient as the variable's trusty sidekick—it's always right next to it, connected by multiplication. In the term $4w$, the coefficient is $4$. To isolate the variable, you need to break up this partnership by dividing both sides of the equation by the coefficient. This makes the coefficient turn into a $1$, leaving the variable by itself.