Lesson 8-3: Solve Linear Equations

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 set up or model a linear equation to fit a real-world application, we must first determine the known quantities and define the unknown quantity as a variable.
Then, we begin to interpret the words as mathematical expressions using mathematical symbols.
For example, a variable cost can be written as $0.10x$, while a fixed cost is a constant added or subtracted, such as in $C = 0.10x + 50$.

To model a linear equation:

1. Identify known quantities.
2. Assign a variable to represent the unknown quantity.
3. If there is more than one unknown quantity, find a way to write the second unknown in terms of the first.
4. Write an equation interpreting the words as mathematical operations.
5. Solve the equation. Be sure the solution can be explained in words, including the units of measure.

Examples

• One number is 10 more than another, and their sum is 52. Let the first number be $x$. The second is $x+10$. The equation is $x + (x+10) = 52$, so $2x=42$, and $x=21$. The numbers are 21 and 31.
• A taxi charges 3 dollars plus 2 dollars per mile. The total cost $C$ for a ride of $x$ miles is modeled by the equation $C = 2x + 3$. A 5-mile ride costs $C = 2(5) + 3 = 13$ dollars.
• Two streaming services have different plans. Plan A is 15 dollars a month. Plan B is 5 dollars a month plus 2 dollars per movie. To find when they cost the same for $m$ movies, set $15 = 5 + 2m$. Solving gives $10 = 2m$, so $m=5$ movies.

Property

Step 1. Choose one side to be the variable side and then the other will be the constant side.
Step 2. Collect the variable terms to the variable side, using the Addition or Subtraction Property of Equality.
Step 3. Collect the constants to the other side, using the Addition or Subtraction Property of Equality.
Step 4. Make the coefficient of the variable 1, using the Multiplication or Division Property of Equality.
Step 5. Check the solution by substituting it into the original equation.

It is a good idea to make the variable side the one in which the variable has the larger coefficient. This usually makes the arithmetic easier.

Examples

• Given $9x + 3 = 4x + 23$, subtract $4x$ from both sides to get $5x + 3 = 23$. Then subtract 3 to get $5x = 20$, so $x = 4$.