Solving One-Step and Two-Step Linear Equations

Hello. In the previous lesson, you evaluated expressions by substituting known values for variables. Solving an equation reverses that perspective: instead of being told the value of a variable, you determine the value that makes an equation true.

In this lesson, you will solve one-step and two-step linear equations by preserving equality, using inverse operations, and checking answers through substitution. These are the basic moves behind nearly all later equation solving.

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Equations as balanced statements

An equation says that two quantities have the same value. The equals sign does not mean “the answer comes next”; it means that the expression on the left has exactly the same value as the expression on the right.

For example,

$$x+3=8$$

states that some number plus 3 has the same value as 8. The goal is to find that number.

A solution is a value of the variable that makes the equation true. Here, $x=5$ is a solution because substituting 5 gives

$$5+3=8,$$

which is true.

Think of the equation as a balanced scale. If you remove, add, multiply, or divide one side, you must make the same change to the other side. Otherwise, the two sides would no longer be equal.

The central principle is called the Properties of Equality:

• Adding the same quantity to both sides preserves equality.
• Subtracting the same quantity from both sides preserves equality.
• Multiplying both sides by the same nonzero quantity preserves equality.
• Dividing both sides by the same nonzero quantity preserves equality.

The target in every problem is to isolate the variable: leave it alone on one side of the equation.

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One-step equations: undo one operation

A one-step equation has one operation separating the variable from being alone. Solve it by applying the operation that undoes it, called its inverse operation.

Operation currently applied to the variable	Inverse operation used to undo it
Add a number	Subtract that number
Subtract a number	Add that number
Multiply by a nonzero number	Divide by that number
Divide by a nonzero number	Multiply by that number

Addition and subtraction equations

Suppose

$$x+16=28.$$

The $+16$ prevents $x$ from standing alone. Subtraction undoes addition, so subtract 16 from both sides:

$$\begin{array}{rl}x+16-16& =28-16\\ x& =12.\end{array}$$

The equation is solved:

$$x=12.$$

The important point is that we did not “move 16 across the equals sign.” We subtracted 16 from both sides. That is why the right side becomes $28-16$.

Now consider subtraction:

$$y-21=43.$$

Since 21 is being subtracted from $y$, add 21 to each side:

$$\begin{array}{rl}y-21+21& =43+21\\ y& =64.\end{array}$$

So,

$$y=64.$$

The sign in the original equation determines the inverse operation:

• To undo $+21$, use $-21$.
• To undo $-21$, use $+21$.

This stays true with decimals and fractions. For example:

$$x+2.7=\mathrm{5.1.}$$

Subtract $2.7$ from both sides:

$$\begin{array}{rl}x+2.7-2.7& =5.1-2.7\\ x& =\mathrm{2.4.}\end{array}$$

The algebraic method is unchanged; only the arithmetic requires more care.

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Multiplication and division equations

A number written directly next to a variable indicates multiplication. Thus,

$$8x=72$$

means

$$8\cdot x=72.$$

Because $x$ is multiplied by 8, divide both sides by 8:

$$\begin{array}{rl}\frac{8x}{8}& =\frac{72}{8}\\ x& =9.\end{array}$$

The fraction on the left simplifies because $8/8=1$, and $1x=x$.

For a division equation such as

$$\frac{x}{25}=6,$$

the variable has been divided by 25. Undo that division by multiplying both sides by 25:

$$\begin{array}{rl}25\left(\frac{x}{25}\right)& =6(25)\\ x& =150.\end{array}$$

A negative coefficient does not change the method. Solve

$$-4x=28.$$

Divide both sides by $-4$:

$$\begin{array}{rl}\frac{-4x}{-4}& =\frac{28}{-4}\\ x& =-7.\end{array}$$

Be especially careful with the sign on the right side. A positive divided by a negative is negative.

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Checking a solution

Checking uses the substitution skill from the previous lesson. Replace the variable with your proposed solution in the original equation, then determine whether both sides have the same value.

For the equation

$$-4x=28,$$

we found

$$x=-7.$$

Substitute:

$$-4(-7)=28.$$

Since

$$28=28,$$

the solution checks.

A check is valuable because it can reveal a sign error, an arithmetic mistake, or an operation performed on only one side. It also clarifies what “solving” means: you have found a value that turns the equation into a true numerical statement.

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Two-step equations: undo operations in reverse order

A two-step equation requires two operations to isolate the variable. Often it has a form such as

$$ax+b=c,$$

where $a$, $b$, and $c$ are known numbers and $a\ne 0$.

To see the order, imagine starting with $x$:

1. Multiply $x$ by $a$.
2. Add $b$.
3. Obtain $c$.

Solving must undo these actions in the reverse order:

1. Undo the addition or subtraction outside the variable term.
2. Undo the multiplication or division attached to the variable.

Consider

$$3x+7=25.$$

The final operation affecting $x$ is addition of 7. First subtract 7 from both sides:

$$\begin{array}{rl}3x+7-7& =25-7\\ 3x& =18.\end{array}$$

Now $x$ is multiplied by 3. Divide both sides by 3:

$$\begin{array}{rl}\frac{3x}{3}& =\frac{18}{3}\\ x& =6.\end{array}$$

Check in the original equation:

$$3(6)+7=25$$ $$18+7=25.$$

The check is true, so

$$x=6.$$

Notice why dividing by 3 first would be unhelpful. The left side is $3x+7$, not simply $3x$. Division would have to apply to the entire left-hand side:

$$\frac{3x+7}{3},$$

which does not immediately isolate $x$. Remove the separate constant term first.

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More two-step structures

The same reverse-order principle works when subtraction, division, or negative coefficients appear.

Example: subtraction and multiplication

Solve

$$-2x+5=17.$$

The $+5$ is outside the variable term, so subtract 5 from both sides:

$$\begin{array}{rl}-2x+5-5& =17-5\\ -2x& =12.\end{array}$$

Now divide by $-2$:

$$\begin{array}{rl}\frac{-2x}{-2}& =\frac{12}{-2}\\ x& =-6.\end{array}$$

Check:

$$-2(-6)+5=17$$ $$12+5=17.$$

Therefore,

$$x=-6.$$

Example: division and subtraction

Solve

$$\frac{x}{4}-3=5.$$

The subtraction of 3 occurs after $x$ is divided by 4, so undo it first by adding 3:

$$\begin{array}{rl}\frac{x}{4}-3+3& =5+3\\ \frac{x}{4}& =8.\end{array}$$

Now multiply both sides by 4:

$$\begin{array}{rl}4\left(\frac{x}{4}\right)& =8(4)\\ x& =32.\end{array}$$

A brief check confirms it:

$$\frac{32}{4}-3=5$$ $$8-3=5.$$

So,

$$x=32.$$

A useful question before each step is: What is currently being done to the variable or variable term? Choose the inverse of that operation, apply it to both sides, and simplify.

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A short application

Equations represent quantities in context, not just symbols on a page. Suppose three identical notebooks cost $18$ dollars in total, and a delivery charge of $4$ dollars is added. The final bill is $58$ dollars. What does one notebook cost?

Let $n$ be the cost, in dollars, of one notebook. The situation translates to

$$3n+4=58.$$

First remove the delivery charge:

$$\begin{array}{rl}3n+4-4& =58-4\\ 3n& =54.\end{array}$$

Then divide the total notebook cost among three notebooks:

$$\begin{array}{rl}\frac{3n}{3}& =\frac{54}{3}\\ n& =18.\end{array}$$

Each notebook costs

$$18\text{ dollars}.$$

Checking against the situation matters:

$$3(18)+4=58.$$

Since $54+4=58$, the answer fits the information given.

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Common errors to avoid

Error	Why it fails	Better habit
Changing only one side of an equation	Equality is no longer preserved.	Write the same operation on both sides.
Treating an equals sign as an instruction to calculate only to the right	An equation is a balanced statement.	Keep both sides visible throughout the solution.
Reversing the wrong operation	The variable does not become more isolated.	Identify the operation immediately outside the variable term.
Dividing only part of a side	Operations on a side apply to the entire side.	Use parentheses or fractions when necessary to show the full side.
Losing a negative sign	A sign error changes the solution.	Put negative numbers in parentheses when substituting to check.
Skipping the check	Small arithmetic errors can survive unnoticed.	Substitute the final value into the original equation.

For this lesson, equations have already been simplified and the variable appears on only one side. In the next lesson, you will extend the same balance principle to equations that require more steps, including simplifying expressions and handling variables on both sides.

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Key takeaways

To solve a one-step or two-step linear equation:

• Treat the equals sign as a statement of balance.
• Isolate the variable by applying inverse operations.
• Perform every operation on both sides of the equation.
• In two-step equations, undo operations in reverse order: remove addition or subtraction outside the variable term before undoing multiplication or division.
• Check your result by substitution in the original equation.

The essential pattern is consistent: preserve equality while progressively leaving the variable alone. Next, you will use this same logic on multi-step equations, where simplifying and collecting variable terms become necessary.