Solving Basic Equations

13 min read

Quick Reference

Try This First

Before any rule, look at this scale.

        ┌───────────────┐         ┌───────────────┐
        │   2x + 3      │   ===   │      11       │
        └───────────────┘         └───────────────┘
              left pan                 right pan

The two pans balance. That is what the equals sign means: whatever sits on the left weighs exactly as much as whatever sits on the right.

Answer two questions in your head, no algebra needed.

1. If you take 3 units of weight off the left pan, what must you do to the right pan to keep it level?
2. After that step, the left pan reads $2x$ and the right pan reads $8$. One copy of $x$ is half of that. What is $x$?

The Balance Idea

An equation is a claim that two expressions name the same number. Solving the equation means finding every input value that makes the claim true.

The picture to keep in mind is a balance scale that starts level. Whatever operation you apply to one side, you apply to the other side, and the scale stays level. Add the same amount to both pans, the balance holds. Multiply both pans by the same nonzero number, the balance holds. Each legal move trades the equation for a simpler equation that has the same solution, until the variable stands alone.

This is a process view. An equation takes a candidate input, and a solution is the input that turns the claim into a true statement. Checking a solution runs that process in reverse to confirm the two sides really do match. The goal of every step is isolation: get the variable alone on one side by applying inverse operations, the operations that undo what was done to the variable.

Prerequisite Hub

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        P["Arithmetic with<br/>signed numbers<br/>and fractions<br/>(review)"]
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        C["Solving Basic<br/>Equations"]
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        D["Algebraic<br/>Simplification"]
        E["Exponent<br/>Laws"]
        F["Linear<br/>Equations"]
        G["Functions and<br/>Function Notation"]
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    click E "exponent-laws.html"
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Builds on: This is a review skill with no listed prerequisites. The reader is assumed comfortable adding, subtracting, multiplying, and dividing signed numbers and fractions.

Unlocks: Mastering the moves on this page is the gate to four later skills.

The Official Definition

Linear equation in one variable. An equation that can be written in the standard form $ax + b = 0$, where $a$ and $b$ are real numbers and $a$ is not equal to $0$.

Source: OpenStax College Algebra 2e, Section 2.2 Linear Equations in One Variable.

Two parts of this definition carry the weight.

First, one variable. A single unknown letter appears, here written $x$. The equation $2x + 3 = 11$ qualifies. An equation with both $x$ and $y$ does not fit this form.

Second, the condition $a \neq 0$. If $a$ were $0$, the term $ax$ would vanish and the statement would read $b = 0$, which carries no variable to solve for. The restriction $a \neq 0$ guarantees the variable is actually present.

The standard form $ax + b = 0$ is a target shape. Many equations do not start in that shape. The work of solving is a sequence of balance moves that brings an equation toward a form where the variable is isolated, and any linear equation in one variable can be rewritten in the standard form above.

The Two Properties of Equality

Every legal move comes from one of two properties. Each one replaces an equation with a simpler equation that has exactly the same solution.

Subtraction is adding a negative, and division is multiplying by a reciprocal, so these two properties cover all four basic moves.

The phrase “nonzero number” in the multiplication property matters. Multiplying both sides by $0$ turns any equation into $0 = 0$, which is true for every input and so erases the information you were trying to keep. Dividing by $0$ is not defined at all.

Order of inverse operations (work from outside in):

1. Undo addition and subtraction first (remove constants from the variable’s side).
2. Undo multiplication and division last (remove the coefficient).

The reason is that an expression like $4x - 3$ was built by multiplying first, then subtracting. To take it apart, reverse that order: add the $3$ first, then divide by the $4$.

Worked Examples

1. One-Step Equations

A one-step equation requires one inverse operation.

Example 1. Solve $x + 9 = 14$.

Predict first. The left side exceeds $x$ by $9$, so $x$ should land $9$ below $14$, near $5$.

The operation done to $x$ is addition of $9$. Subtract $9$ from both sides: \[ x + 9 - 9 = 14 - 9 \] \[ x = 5 \]

Check: $5 + 9 = 14$. The prediction holds. $\checkmark$

Example 2. Solve $7x = 42$.

Predict first. Seven copies of $x$ make $42$, so one copy is about $42 \div 7$, near $6$.

The operation done to $x$ is multiplication by $7$. Divide both sides by $7$: \[ \frac{7x}{7} = \frac{42}{7} \] \[ x = 6 \]

Check: $7(6) = 42$. $\checkmark$

Example 3. Solve $\dfrac{x}{5} = -3$.

Predict first. A fifth of $x$ is negative, so $x$ itself is negative and five times as large, near $-15$.

Multiply both sides by $5$: \[ x = -15 \]

Check: $\dfrac{-15}{5} = -3$. $\checkmark$

2. Two-Step Equations

A two-step equation requires two inverse operations. Undo addition and subtraction first, then multiplication and division.

Example 4. Solve $4x - 3 = 13$.

Predict first. The left side grows as $x$ grows. At $x = 4$ the left side reads $4(4) - 3 = 13$, so the answer should be near $4$.

Step 1. Add $3$ to both sides (undo subtraction): \[ 4x = 16 \]

Step 2. Divide by $4$ (undo multiplication): \[ x = 4 \]

Check: $4(4) - 3 = 16 - 3 = 13$. $\checkmark$

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

Predict first. Half of $x$ plus $7$ equals $1$, so half of $x$ is about $-6$, and $x$ is about $-12$.

Step 1. Subtract $7$ from both sides: \[ \frac{x}{2} = -6 \]

Step 2. Multiply both sides by $2$: \[ x = -12 \]

Check: $\dfrac{-12}{2} + 7 = -6 + 7 = 1$. $\checkmark$

3. Variable on Both Sides

When the variable appears on both sides, gather the variable terms on one side and the constants on the other.

Example 6. Solve $5x - 4 = 2x + 11$.

Predict first. The left side has the larger coefficient on $x$, so it catches up to and passes the right side as $x$ grows. The solution is the single input where they are equal, in the positive range.

Subtract $2x$ from both sides: \[ 5x - 4 - 2x = 2x + 11 - 2x \] \[ 3x - 4 = 11 \]

Add $4$ to both sides: \[ 3x = 15 \]

Divide both sides by $3$: \[ x = 5 \]

Check: Left side $5(5) - 4 = 21$. Right side $2(5) + 11 = 21$. Both read $21$. $\checkmark$

4. Negative and Fractional Coefficients

Dividing by a negative number is valid; the result may change sign.

Example 7. Solve $-8x = 56$.

Predict first. A negative coefficient times $x$ gives a positive result, so $x$ must be negative, near $-7$.

Divide by $-8$: \[ x = \frac{56}{-8} = -7 \]

Check: $-8(-7) = 56$. $\checkmark$

Important: Multiplying or dividing both sides of an equation by a negative number is fine. This differs from an inequality, where doing so requires flipping the inequality sign.

For a fractional coefficient, multiply both sides by the reciprocal of the coefficient.

Example 8. Solve $\dfrac{3}{4}x = 9$.

Multiply both sides by $\dfrac{4}{3}$ (the reciprocal of $\dfrac{3}{4}$): \[ x = 9 \cdot \frac{4}{3} = 12 \]

Check: $\dfrac{3}{4}(12) = 9$. $\checkmark$

5. Equations with Parentheses and Standard Form

When more than two operations are present, apply inverse operations in sequence, working from outside to inside.

Example 9. Solve $4(x - 1) = x + 5$, then write the result in the standard form $ax + b = 0$.

Distribute on the left: \[ 4x - 4 = x + 5 \]

Subtract $x$ from both sides: \[ 3x - 4 = 5 \]

Add $4$ to both sides: \[ 3x = 9 \]

Divide both sides by $3$: \[ x = 3 \]

Standard form. Moving every term in $3x - 4 = 5$ to the left gives $3x - 9 = 0$, which matches $ax + b = 0$ with $a = 3$ and $b = -9$. Because $a = 3 \neq 0$, this is a genuine linear equation in one variable.

Check: Left side $4(3 - 1) = 4(2) = 8$. Right side $3 + 5 = 8$. Both read $8$. $\checkmark$

Two Ways to See the Same Problem

The equation $5x - 4 = 2x + 11$ from Example 6 has more than one valid path.

One way to see this: Move the variable terms to the left. Subtracting $2x$ from both sides gives $3x - 4 = 11$, then $x = 5$.

Another way to see this: Move the variable terms to the right. Subtracting $5x$ from both sides gives $-4 = -3x + 11$, then $-15 = -3x$, then $x = 5$.

Both routes reach the same solution because every balance move preserves the solution. Which route do you prefer, and why? Many readers keep the variable on the side with the larger coefficient to avoid a negative coefficient, yet either path is correct.

Common Misconceptions

Why Checking Is Not Optional

Checking a solution is the only way to be certain an answer is correct.

1. Arithmetic slips are common. Substituting back catches them at once.
2. Some equation types (especially those with square roots or denominators) can produce extraneous solutions, values that satisfy the algebra but not the original equation.
3. The check reinforces what “solution” means: the value that makes the equation true.

To check, substitute the value into the original, unmodified equation and confirm both sides equal the same number.

Example 10. A student solved $5x - 3 = 22$ and reported $x = 4$. Check: $5(4) - 3 = 20 - 3 = 17 \neq 22$. The value fails, and the failure points to the step where the arithmetic went wrong. The correct solution is $x = 5$, since $5(5) - 3 = 22$.

Common Errors Summary

Leveled Practice

Ask Why

Why does dividing both sides by the same nonzero number leave the solution unchanged, while dividing by zero is forbidden? Try to convince a classmate in one or two sentences using the balance picture, then compare your explanation with the note in the Two Properties of Equality section.

A wrong turn anywhere on this page is common and shows something worth understanding. There is no need to rush. When a check fails, the failure points straight at the step where one pan changed without the other, which is exactly the place to look.

Mastery Checklist

Novice (Level 1-2):

• I can state what the equals sign claims and what it means to solve an equation
• I can identify the operation done to the variable and apply its inverse
• I can solve a one-step or two-step equation and check the result by substitution

Competent (Level 3-4):

• I can solve an equation with the variable on both sides by gathering like terms
• I can solve equations with negative and fractional coefficients
• I can distribute or divide first to handle parentheses, and explain why both paths work

Proficient (Level 5):

• I can rewrite a solved equation in the standard form $ax + b = 0$ and identify $a$ and $b$
• I can explain why the restriction $a \neq 0$ is part of the definition
• I can describe when a linear equation in one variable has one solution or none

Mental Model

The “Balance Scale” Analogy:

Picture every equation as a scale that starts level. The equals sign is the pivot. Each legal step changes both pans the same way, so the scale never tips. Solving is a series of such steps that strips away everything piled on top of the variable until a single $x$ stands alone in one pan and its value rests in the other.

Another way to picture it is a recipe in reverse. Someone took a number, performed operations on it in a fixed order, and arrived at a result. You see only the result and the operations. Reversing the recipe step by step, undoing the last operation first, recovers the original number.

Connections

Looking back:

• The arithmetic of signed numbers and fractions feeds directly into every balance move.

Looking ahead:

• Algebraic Simplification: combining like terms before solving
• Exponent Laws: equations involving powers reuse the balance moves
• Linear Equations: the full study of $ax + b = c$, slopes, and lines
• Functions and Function Notation: solving for the input is the same reasoning as evaluating and inverting a function

Real-world connections:

• A budget that must balance income against fixed and variable costs is a linear equation in the unknown quantity
• Physical formulas such as Ohm’s law $V = IR$ and Newton’s second law $F = ma$ become one-step or two-step equations when one quantity is unknown
• Unit conversions and break-even calculations all reduce to isolating a single unknown

Resources

• Primary text: OpenStax College Algebra 2e, Section 2.2 Linear Equations in One Variable. https://openstax.org/books/college-algebra-2e/pages/2-2-linear-equations-in-one-variable
• Chapter summary: OpenStax College Algebra 2e, Chapter 2 Key Concepts. https://openstax.org/books/college-algebra-2e/pages/2-key-concepts

Last updated: 2026-06-16