2.1 Use a General Strategy to Solve Linear Equations

Solving Linear Equations

Linear equations are the building blocks of algebra. They show up constantly in later math courses, so getting comfortable solving them now will pay off. This section covers a general strategy that works for any linear equation, including ones with fractions, decimals, and variables on both sides.

Steps for Solving Linear Equations

The same general strategy applies to every linear equation. Here it is as a step-by-step process:

1. Simplify each side of the equation separately.

   • Use the distributive property to clear parentheses: $2(3x + 4) = 6x + 8$
   • Combine like terms on each side: $3x + 2x = 5x$

2. Collect variable terms on one side and constants on the other.

   • Add or subtract terms from both sides to move all variable terms to one side and all constant terms to the other. For example, given $2x - 3 = 7$, add 3 to both sides to get $2x = 10$.

3. Divide both sides by the coefficient of the variable.

   • $2x = 10$ becomes $x = 5$ after dividing both sides by 2.

4. Check your solution by substituting it back into the original equation. If both sides are equal, you're good.

This strategy works whether the equation is simple or messy. The key is always simplify first, then isolate the variable.

Types of Linear Equations

Not every linear equation has exactly one answer. There are three possible outcomes:

• One solution: The equation simplifies so you can isolate the variable and get a single value. For example, $2x - 3 = 7$ gives $x = 5$.
• No solution (contradiction): The equation simplifies to a false statement like $0 = 1$. This means no value of $x$ will ever make the equation true. For example, $3x + 1 = 3x + 5$ simplifies to $1 = 5$, which is false.
• Infinitely many solutions (identity): The equation simplifies to a statement that's always true, like $0 = 0$. Every real number is a solution. For example, $2(x + 3) = 2x + 6$ simplifies to $6 = 6$.

Equations with variables on both sides require an extra step: collect all the variable terms on one side first. For instance, with $2x + 3 = x + 7$, subtract $x$ from both sides to get $x + 3 = 7$, then subtract 3 to get $x = 4$.

Equations with parentheses just need the distributive property applied before you proceed with the general strategy. For $2(3x + 4) = 10$, distribute to get $6x + 8 = 10$, then solve as usual: subtract 8, then divide by 6 to get $x = \frac{1}{3}$.

Solving Equations with Fractions and Decimals

Fractions and decimals make equations look harder, but there's a clean trick: clear them out first by multiplying every term by the least common denominator.

1. Find the LCD (least common multiple of all denominators in the equation).
2. Multiply every term on both sides by that LCD. This eliminates all fractions in one step.
3. Solve the resulting equation using the general strategy above.
4. Check your answer by plugging it back into the original equation (the one with fractions).

Example: Solve $\frac{x}{2} + \frac{x}{3} = 5$

• The LCD of 2 and 3 is 6.
• Multiply every term by 6: $6 \cdot \frac{x}{2} + 6 \cdot \frac{x}{3} = 6 \cdot 5$, which gives $3x + 2x = 30$.
• Combine like terms: $5x = 30$.
• Divide by 5: $x = 6$.

For decimals, you can multiply both sides by a power of 10 to clear the decimals. For instance, if every decimal goes to the tenths place, multiply everything by 10. If some go to the hundredths place, multiply by 100. Then solve normally.

Understanding Equivalent Equations

Two equations are equivalent if they have the exact same solution set. Every time you add, subtract, multiply, or divide both sides of an equation by the same nonzero value, you produce an equivalent equation.

This is actually the principle behind the entire solving strategy. Each step transforms the equation into a simpler equivalent equation, until you reach one that directly tells you the value of the variable. That's why performing the same operation on both sides is always valid: it changes the equation's appearance but never changes its solution.