Lesson 7: Solving Linear Equations (Exploration: Solving Equations Using Algebra Tiles)

New Concept

An equation is a statement which indicates that two expressions are equal.

Why it matters

Equations are the language used to describe relationships in everything from physics to finance. Mastering the properties of equality is your first step toward building complex models and solving for unknown variables in any system.

What’s next

Next, you’ll apply the properties of equality to solve equations with variables on one or both sides, and even those involving fractions.

A number is a solution of an equation in one variable if substituting the number for the variable results in a true statement.

Is $x = 5$ a solution to $3x - 2 = 13$? Check: $3(5) - 2 = 15 - 2 = 13$. Yes, it's a true statement!|Is $y = -2$ a solution to $4y + 10 = 3$? Check: $4(-2) + 10 = -8 + 10 = 2$. No, because $2 \neq 3$.|To confirm the solution to $6n+1=13$ is $n=2$, substitute it back: $6(2)+1 = 12+1 = 13$. It checks out!

Finding a solution is like discovering the secret code that opens a lock! You are hunting for that one specific number that you can substitute for the variable to make both sides of the equation perfectly equal. If they match after you plug it in, you have successfully found the code. If they do not, the mystery remains unsolved.

Addition: If $a = b$ then $a + c = b + c$. Subtraction: If $a = b$ then $a - c = b - c$. Multiplication: If $a = b$ then $ac = bc$. Division: If $a = b$ and $c \neq 0$ then $\frac{a}{c} = \frac{b}{c}$.

To solve $x - 8 = 10$, use the Addition Property: $x - 8 + 8 = 10 + 8$, which simplifies to $x = 18$.
To solve $4y = 24$, use the Division Property: $\frac{4y}{4} = \frac{24}{4}$, which simplifies to $y = 6$.
To solve $-2x + 5 = 15$, first subtract 5: $-2x = 10$. Then divide by -2: $x = -5$.

Imagine an equation is a perfectly balanced seesaw. To keep it level, whatever you do to one side, you must do the exact same thing to the other. Whether you add, subtract, multiply, or divide by a number, applying the action to both sides ensures the equation remains true, helping you isolate the variable and solve the puzzle.

To solve equations with variables on both sides, first use the Addition or Subtraction Property of Equality to collect the variable terms on one side of the equation. Then, use the properties of equality to isolate the variable.

Solve $5x - 4 = 2x + 11$. First, subtract $2x$ from both sides to get $3x - 4 = 11$.|Continuing from $3x - 4 = 11$, add 4 to both sides: $3x = 15$. Now, divide by 3 to get $x = 5$.|For $8y + 5 = 10y - 1$, subtract $8y$ from both sides: $5 = 2y - 1$. Then add 1: $6 = 2y$, so $y = 3$.

When variables appear on both sides of an equation, it is like a mathematical tug-of-war. Your first move is to gather all the variable terms onto one team by adding or subtracting them from both sides. Once all variables are grouped together, you can combine them and use the properties of equality to find out who wins!