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One-variable equations

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Understanding One-Variable Equations

In this chapter we focus on equations that contain just one variable, usually written as $x$ or another single letter. The goal is always the same: find which value (or values) of the variable make the equation true.

An example of a one-variable equation is
$$
3x + 5 = 20.
$$
Here there is only one variable, $x$.

You will see many different forms of such equations, but they all share a basic idea: two expressions are equal, and you are allowed to perform certain operations on both sides to uncover the value of the variable.

The Idea of “Solving” an Equation

To solve a one-variable equation means:

For example, in $x + 7 = 10$, we are looking for a number that, when we add 7, gives 10. That number is $3$, so we say $x = 3$ is the solution.

You can always check a proposed solution by substituting it back into the original equation and seeing if both sides match:

They are equal, so $x=3$ is correct.

Basic Strategy: “Undo” Operations

Most one-variable equations in pre-algebra can be solved by undoing the operations that are applied to the variable. You work backwards, using inverse operations:

The usual process:

  1. Identify the variable term (the part containing $x$).
  2. Undo additions or subtractions.
  3. Undo multiplications or divisions.
  4. Check your answer by substitution.

You must always do the same operation to both sides of the equation to keep it balanced.

Using Inverse Operations

Undoing addition or subtraction

If $x$ has a number added to it, you subtract that number from both sides.

Example:
$$
x + 4 = 9
$$
Subtract $4$ from both sides:
$$
x + 4 - 4 = 9 - 4
$$
So
$$
x = 5.
$$

If $x$ has a number subtracted from it, you add that number to both sides.

Example:
$$
x - 6 = 2
$$
Add $6$ to both sides:
$$
x - 6 + 6 = 2 + 6
$$
So
$$
x = 8.
$$

Undoing multiplication or division

If the variable is multiplied by a number, you divide both sides by that number.

Example:
$$
5x = 20
$$
Divide both sides by $5$:
$$
\frac{5x}{5} = \frac{20}{5}
$$
So
$$
x = 4.
$$

If the variable is divided by a number, you multiply both sides by that number.

Example:
$$
\frac{x}{3} = 7
$$
Multiply both sides by $3$:
$$
3 \cdot \frac{x}{3} = 7 \cdot 3
$$
So
$$
x = 21.
$$

Two-Step Equations

Many one-variable equations require two steps to undo the operations.

Example:
$$
3x + 2 = 14
$$

  1. Undo the addition or subtraction first:
    $$
    3x + 2 - 2 = 14 - 2
    $$
    So
    $$
    3x = 12.
    $$
  2. Then undo the multiplication or division:
    $$
    \frac{3x}{3} = \frac{12}{3}
    $$
    So
    $$
    x = 4.
    $$

Check:

So $x = 4$ is correct.

The general pattern for equations of the form
$$
ax + b = c
$$
(with $a \neq 0$) is:

  1. Subtract $b$ from both sides: $ax = c - b$.
  2. Divide both sides by $a$: $x = \dfrac{c - b}{a}$.

Equations with Subtraction Written in Different Ways

Sometimes subtraction appears in a slightly different form, but can be treated similarly.

Example: $7 - x = 3$

Here the variable is being subtracted from $7$. To solve, you can isolate the term with $x$.

One method is to subtract $7$ from both sides:
$$
7 - x - 7 = 3 - 7
$$
So
$$
-x = -4.
$$
Now $-x$ means $-1 \cdot x$. To undo this multiplication, divide both sides by $-1$ (or multiply by $-1$):
$$
x = 4.
$$

Check:

So $x = 4$ works.

The important idea is that if you end up with $-x$ on one side, you can multiply both sides by $-1$ to change it to $x$.

Equations with the Variable on Both Sides

Sometimes $x$ appears on both sides of the equation. The key idea is to collect all the $x$-terms on one side and all the constant numbers on the other side.

Example:
$$
4x + 5 = 2x + 13
$$

  1. Get all $x$ terms on one side. For example, subtract $2x$ from both sides:
    $$
    4x + 5 - 2x = 2x + 13 - 2x
    $$
    So
    $$
    2x + 5 = 13.
    $$
  2. Now solve this two-step equation. Subtract $5$ from both sides:
    $$
    2x = 8.
    $$
    Divide by $:
    $$
    x = 4.
    $$

Check:

So $x=4$ is correct.

You could also choose to subtract $4x$ from both sides; either way is valid. Choosing the smaller $x$-coefficient to subtract often keeps the numbers simpler, but it is not required.

No Solution and Infinitely Many Solutions

Not every one-variable equation has exactly one solution. When you simplify, you might end up with:

Case 1: No solution

Example:
$$
2x + 3 = 2x - 5
$$

Subtract $2x$ from both sides:
$$
2x + 3 - 2x = 2x - 5 - 2x
$$
So
$$
3 = -5.
$$

This is impossible. There is no value of $x$ that can make $3 = -5$ true, so we say the equation has no solution.

Sometimes this is written as “no solution” or with a special symbol $\varnothing$ (the empty set).

Case 2: Infinitely many solutions

Example:
$$
3(x + 2) = 3x + 6
$$

Distribute on the left (expanding the parentheses is covered elsewhere, so just note the result):
$$
3x + 6 = 3x + 6.
$$

Subtract $3x$ from both sides:
$$
6 = 6.
$$

This is always true, no matter what $x$ is. The equation is true for every real number $x$. We say it has infinitely many solutions, or the solution set is “all real numbers.”

Checking Your Work

For one-variable equations, checking is straightforward:

  1. Take the value you found for $x$.
  2. Substitute it into the original equation.
  3. Simplify both sides.
  4. If both sides are equal, your solution is correct; if not, there was a mistake.

Example:
Solve $5x - 1 = 19$.

Add $1$ to both sides:
$$
5x = 20.
$$
Divide by $5$:
$$
x = 4.
$$

Check in the original equation:

They match, so $x = 4$ is correct.

Checking is especially useful when equations become more complicated (for example, when fractions or parentheses appear). It helps catch arithmetic or sign errors.

Solving Equations with Fractions on One or Both Sides

When equations include fractions, the ideas are the same, but the arithmetic may be a bit more involved. Often you will see:

Simple fraction with $x$ in the numerator

Example:
$$
\frac{x}{4} + 2 = 5
$$

  1. Subtract $2$ from both sides:
    $$
    \frac{x}{4} = 3.
    $$
  2. Multiply both sides by $4$:
    $$
    x = 12.
    $$

Check:
$$
\frac{12}{4} + 2 = 3 + 2 = 5.
$$

Fraction coefficient in front of $x$

Example:
$$
\frac{2}{5}x = 6
$$

To undo multiplying by $\frac{2}{5}$, multiply both sides by its reciprocal, $\frac{5}{2}$:
$$
\frac{5}{2} \cdot \frac{2}{5}x = \frac{5}{2} \cdot 6
$$
So
$$
x = \frac{5}{2} \cdot 6 = \frac{5}{2} \cdot \frac{6}{1} = \frac{30}{2} = 15.
$$

Clearing fractions (idea)

If an equation has several fractions, one common method (in later work) is to multiply both sides by a common denominator to “clear” the fractions. The details of finding and using a common denominator are treated more fully elsewhere, but the main idea is that you can use the same multiplication on both sides of the equation to remove denominators and make the equation easier to solve.

Common Mistakes to Watch For

When solving one-variable equations, several errors appear frequently:

Developing a habit of checking your final answer in the original equation is one of the best ways to catch these mistakes early.

Summary

One-variable equations ask you to find values of a single variable that make two expressions equal. You solve them by:

Mastering these techniques prepares you for word problems involving equations and for more advanced algebraic equations later on.

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