Lesson 2: Solving Linear Equations

Property

A linear equation is an equation in one variable that can be written as $ax + b = 0$, where $a$ and $b$ are real numbers and $a \neq 0$.
To solve linear equations using a general strategy:
Step 1. Simplify each side of the equation as much as possible by using the Distributive Property and combining like terms.
Step 2. Collect all variable terms on one side of the equation.
Step 3. Collect all constant terms on the other side.
Step 4. Make the coefficient of the variable term equal to 1.
Step 5. Check the solution by substituting it into the original equation.

Examples

• Solve $5(x-3) + 1 = -9$. First, distribute: $5x - 15 + 1 = -9$. Combine terms: $5x - 14 = -9$. Add 14 to both sides: $5x = 5$. Divide by 5 to get $x=1$.
• Solve $8k - 5 = 2k + 13$. Subtract $2k$ from both sides: $6k - 5 = 13$. Add 5 to both sides: $6k = 18$. Divide by 6 to get $k=3$.
• Solve $-(z+4) = 10$. Distribute the negative sign: $-z - 4 = 10$. Add 4 to both sides: $-z = 14$. Multiply by $-1$ to get $z=-14$.

Explanation

The main goal is to isolate the variable. Think of it as unwrapping a gift: use inverse operations in reverse order to get the variable by itself. Always simplify both sides first to make the process easier.

Property

To solve equations with fraction or decimal coefficients:
Step 1. Find the least common denominator (LCD) of all the fractions and decimals (in fraction form) in the equation.
Step 2. Multiply both sides of the equation by that LCD. This clears the fractions and decimals.
Step 3. Solve using the General Strategy for Solving Linear Equations.

Examples

• Solve $\frac{1}{5}x - 1 = \frac{3}{5}$. The LCD is 5. Multiply all terms by 5: $5(\frac{1}{5}x) - 5(1) = 5(\frac{3}{5})$, which simplifies to $x - 5 = 3$. The solution is $x=8$.
• Solve $0.2y + 0.05 = 0.45$. To clear decimals, multiply by 100: $100(0.2y) + 100(0.05) = 100(0.45)$, which gives $20y + 5 = 45$. This simplifies to $20y = 40$, so $y=2$.
• Solve $\frac{1}{2}(z+1) = \frac{2}{3}$. The LCD is 6. Multiply both sides by 6: $6 \cdot \frac{1}{2}(z+1) = 6 \cdot \frac{2}{3}$, which gives $3(z+1) = 4$. So, $3z+3=4$, which means $3z=1$, and $z=\frac{1}{3}$.

Explanation

Fractions and decimals can be messy. To simplify your work, find the least common denominator (LCD) of all terms and multiply both sides of the equation by it. This clears the fractions, leaving a simpler integer equation to solve.