Solving Linear Equations in One Variable

8 min read

Textbook Reference

This source is free and openly licensed. College Algebra 2e Section 2.2 covers linear equations, including those with fractions.

Key idea

An equation is a balance scale, and solving it is the act of stripping away everything that surrounds the variable while keeping the scale balanced.

The two sides of an equation weigh the same. Whatever you do to one side you must do to the other, or the balance breaks. To find the value of $x$, you peel off the operations wrapped around it one at a time, always undoing the outermost operation first. This is the order of operations run in reverse: you built the expression by following PEMDAS, so you unbuild it by going the opposite direction.

This means: solving is unwrapping. Identify what is being done to $x$, then undo it in reverse order, doing the same thing to both sides at every step. The goal is to leave $x$ standing alone.

Prerequisite Check

Before this lesson, make sure you can do all of the following:

• Simplify expressions by distributing and combining like terms
• Apply the order of operations
• Add, subtract, multiply, and divide real numbers, including negatives
• Add and subtract fractions with a common denominator

If simplifying is shaky, review Simplifying Algebraic Expressions first.

Quick Reference

A linear equation in one variable can be written in the form \[ ax + b = 0 \quad \text{where } a \neq 0. \] Its solution is the single value of $x$ that makes the equation true.

The properties of equality (the legal moves). For any real numbers, if $A = B$ then:

The solving procedure (flashcard).

1. Clear fractions by multiplying every term by the common denominator (optional but tidy).
2. Distribute to remove parentheses.
3. Combine like terms on each side.
4. Move all variable terms to one side, all constants to the other (add or subtract).
5. Divide by the coefficient of the variable.
6. Check by substituting back into the original equation.

Key Concepts

1. What an Equation Is

An equation states that two expressions are equal. A solution is a value of the variable that makes the statement true. Solving means finding every such value.

Example 1. Verify that $x = 4$ is a solution of $2x + 1 = 9$.

Substitute: $2(4) + 1 = 8 + 1 = 9$. The left side equals the right side, so $x = 4$ is a solution.

A linear equation in one variable has exactly one solution (unless it is a special case, covered in Section 5).

2. Undoing One Operation at a Time

To isolate the variable, undo the operations wrapped around it, doing the same thing to both sides. Undo the outermost operation first.

Example 2. Solve $x + 7 = 12$.

The variable has $7$ added to it. Undo by subtracting $7$ from both sides: \[ x + 7 - 7 = 12 - 7 \implies x = 5 \]

Example 3. Solve $3x = 21$.

The variable is multiplied by $3$. Undo by dividing both sides by $3$: \[ \frac{3x}{3} = \frac{21}{3} \implies x = 7 \]

Important: whatever you do to one side, do to the other. Subtracting $7$ from only the left side breaks the balance and gives a wrong answer. The equality is preserved only when both sides change identically.

3. Two-Step Equations

When more than one operation surrounds the variable, undo them in reverse order: undo addition or subtraction before multiplication or division.

Example 4. Solve $2x + 5 = 17$.

Step 1. Undo the $+5$: subtract $5$ from both sides. $2x = 12$.

Step 2. Undo the $\times 2$: divide both sides by $2$. $x = 6$.

Check: $2(6) + 5 = 12 + 5 = 17$. Correct.

Important: undo addition before division. A frequent error is dividing $2x + 5 = 17$ by $2$ first, writing $x + 5 = 8.5$. The $+5$ is not inside the multiplication, so it must be removed first. Reverse PEMDAS: subtraction before division.

4. Variables on Both Sides

When the variable appears on both sides, first move all variable terms to one side and all constants to the other.

Example 5. Solve $5x - 3 = 2x + 9$.

Step 1. Subtract $2x$ from both sides to gather variables on the left: $3x - 3 = 9$.

Step 2. Add $3$ to both sides: $3x = 12$.

Step 3. Divide by $3$: $x = 4$.

Check: Left $5(4) - 3 = 17$. Right $2(4) + 9 = 17$. Correct.

5. Equations with Parentheses and Fractions

Distribute to clear parentheses, and multiply through by a common denominator to clear fractions, before solving.

Example 6. Solve $3(x - 2) = 2x + 4$.

Distribute: $3x - 6 = 2x + 4$. Subtract $2x$: $x - 6 = 4$. Add $6$: $x = 10$.

Check: $3(10 - 2) = 3(8) = 24$ and $2(10) + 4 = 24$. Correct.

Example 7. Solve $\dfrac{x}{2} + \dfrac{1}{3} = \dfrac{5}{6}$.

The common denominator is $6$. Multiply every term by $6$: \[ 6 \cdot \frac{x}{2} + 6 \cdot \frac{1}{3} = 6 \cdot \frac{5}{6} \implies 3x + 2 = 5 \] Subtract $2$: $3x = 3$. Divide by $3$: $x = 1$.

Check: $\dfrac{1}{2} + \dfrac{1}{3} = \dfrac{3}{6} + \dfrac{2}{6} = \dfrac{5}{6}$. Correct.

6. Special Cases: No Solution and All Reals

Not every linear equation has exactly one solution. After simplifying, two special outcomes can appear.

• If the variable cancels and a false statement remains (such as $3 = 7$), the equation has no solution.
• If the variable cancels and a true statement remains (such as $5 = 5$), every real number is a solution; this is an identity.

Example 8. Solve $2(x + 3) = 2x + 6$.

Distribute: $2x + 6 = 2x + 6$. Subtract $2x$: $6 = 6$. This is always true, so the equation is an identity and every real number is a solution.

Example 9. Solve $x + 4 = x + 9$.

Subtract $x$: $4 = 9$. This is false, so the equation has no solution.

Common Errors Summary

Common Misconceptions

Leveled Practice

Level 1 -- Direct Application

Problem 1. Solve $x - 8 = 3$.

Problem 2. Solve $4x = 28$.

Problem 3. Solve $3x - 5 = 16$.

Level 2 -- Multiple Steps

Problem 4. Solve $7x + 2 = 4x + 17$.

Problem 5. Solve $2(3x - 4) = 4x + 6$.

Level 3 -- Fractions and Special Cases

Problem 6. Solve $\dfrac{x}{3} - \dfrac{1}{2} = \dfrac{x}{6}$.

Problem 7. Solve $3(x + 2) = 3x + 5$ and interpret the result.

Mastery Checklist

You have mastered this skill when you can do all of the following without referring to notes:

• State that a linear equation in one variable has the form $ax + b = 0$ with $a \neq 0$
• Use the properties of equality to undo operations, doing the same to both sides
• Solve one-step and two-step equations by reversing the order of operations
• Gather variable terms on one side when the variable appears on both sides
• Clear parentheses and fractions before solving
• Recognize identities (all reals) and contradictions (no solution)
• Check a solution by substituting it into the original equation

Mental Model

Think of an equation as a balance scale and the variable as a gift buried inside layers of wrapping paper.

The two pans of the scale weigh the same. To find the gift you remove wrapping one layer at a time, and to keep the scale level you must remove the same weight from both pans at every step. You unwrap in the reverse of the order the gift was wrapped: the order of operations built the expression, so you peel it off in reverse, additions and subtractions before multiplications and divisions. When the last layer comes off, the variable stands alone and its value is revealed. If the variable disappears entirely, the scale is telling you something special: a true leftover means every number balances (an identity), and a false leftover means nothing balances (no solution).

Connections

Within MATH 114

• Functions: finding where a function equals a particular output, $f(x) = k$, is solving a linear (or later, a quadratic) equation.
• Applications and variation: word problems translate a real situation into an equation, which you then solve with this procedure.

Toward Later Mathematics

• Systems of equations: solving two equations together builds directly on solving one.
• Calculus (MATH 161): setting a derivative equal to zero to find a critical point produces an equation to solve; the algebra here is the bottleneck skill there.

Audience Notes

For students who find math intimidating: There is one idea to hold onto: do the same thing to both sides, and undo the outer operation first. Every problem is the same dance with different numbers. Checking your answer at the end turns “I hope this is right” into “I know this is right”.

For students interested in proof: Each step is justified by a named property of equality. Writing the property beside each line turns a calculation into a short proof that the final value really is the solution.

For students interested in careers: Solving for an unknown is the most common quantitative task in any technical field: rearranging a formula to isolate the quantity you need is exactly this skill, applied to letters instead of a lone $x$.

Back to MATH 114 Skills | Previous: Simplifying Algebraic Expressions | Next: The Rectangular Coordinate Plane