How to Solve Linear Equations

Algebra is arithmetic with a blank space in it. A letter such as x stands for a number you have not found yet, and solving means finding the number that makes the sentence true.

How to Solve Linear Equations: the central idea

A linear equation asks for every value that makes two first-degree expressions equal. Solving preserves that equality while isolating the variable.

Words you need

Linear equation

A linear equation is an equation in which the variable appears only to the first power, so its graph is a straight line.

Coefficient

A coefficient is the number multiplied by a variable, such as the 3 in $3x$.

Like terms

Like terms are terms with exactly the same variable part, such as $5x$ and $-2x$, so they can be combined into one term.

Inverse operation

An inverse operation is the operation that undoes another one, so subtraction undoes addition and division undoes multiplication.

Identity

An identity is an equation that stays true no matter what number you substitute, such as $5(x-2)=5x-10$.

Solution set

A solution set is the collection of every value that makes the equation true, and it may hold one number, no numbers at all, or all real numbers.

What to know before this lesson

Be comfortable with signed-number arithmetic, the distributive property, and combining like terms.

If one of those prerequisites is uncertain, use the Algebra subject guide to locate the earlier concept before memorizing a procedure.

Linear equations: a worked example

Every step, with the arithmetic

1. Step 1 - Write the equation$4(x+3)-2x=3x-5$
2. Step 2 - Distribute the 4 across the parentheses$4x+12-2x=3x-5$
3. Step 3 - Combine the like terms on the left$2x+12=3x-5$
4. Step 4 - Subtract $2x$ from both sides$12=x-5$
5. Step 5 - Add 5 to both sides$17=x$
6. Step 6 - Check the left side with $x=17$$4(17+3)-2(17)=80-34=46$
7. Step 7 - Check the right side with $x=17$$3(17)-5=51-5=46$, and $46=46$, so $x=17$ is correct

In $3x-5=16$, adding $5$ removes the constant attached to the variable side. Dividing $21$ by $3$ then reverses multiplication and produces $x=7$.

The three things a linear equation can do

The step-by-step method for linear equations

1. Expand parentheses and combine like terms on each side without crossing the equals sign.
2. Move variable terms to one side and constants to the other using the same operation on both sides.
3. Divide by the remaining coefficient, then state whether the result is one solution, no solution, or every real number.

How to check your answer

Substitute $7$ into the untouched equation: $3(7)-5=21-5=16$. Because both sides match, the value belongs to the solution set.

Substitute a proposed value into the original statement—not only the last simplified line. For functions, also inspect domain, intercepts, and whether the graph agrees.

A mistake that changes the mathematics

Moving a term does not magically change its sign. The sign changes because you add or subtract the same expression on both sides.

Where you will actually use this

Finding a break-even point

A stand spends 500 dollars on equipment plus 3 dollars per drink, and sells each drink for 8 dollars. Setting cost equal to income gives $500+3x=8x$, so $500=5x$ and $x=100$ drinks. Below 100 drinks the stand loses money; above it, the stand earns money.

Converting a temperature

Fahrenheit and Celsius are linked by $F=\tfrac95C+32$. To find the Celsius reading for 68 degrees Fahrenheit, solve $68=\tfrac95C+32$: subtract 32 to get $36=\tfrac95C$, then multiply by $\tfrac59$ to get $C=20$.

Splitting a shared bill

Four friends split a dinner bill evenly after a 10 dollar coupon, and each pays 17 dollars. If $b$ is the bill, then $(b-10)/4=17$, so $b-10=68$ and $b=78$ dollars.

How linear equations connects to the rest of algebra

• Systems of equations — A system is two linear equations at once, and both substitution and elimination end with a single linear equation solved exactly the way you solve one here.
• The quadratic formula — Once the variable is squared the equation stops being linear, and moving terms around is no longer enough to finish it.
• Order of operations — Solving runs the order of operations backwards: you peel off addition and subtraction first and undo the multiplication last.

Try a transfer problem

Solve $5(2x-1)=3x+16$, then explain which operation preserves equivalence at every line.

Show the worked answer

Distribute on the left: $5(2x-1)=10x-5$, so the equation is $10x-5=3x+16$. Subtract $3x$ from both sides: $7x-5=16$. Add 5 to both sides: $7x=21$. Divide both sides by 7: $x=3$. Check in the original equation: $5(2\cdot3-1)=5(5)=25$ and $3(3)+16=25$, so both sides agree. Each line kept the same solution for a specific reason. Distributing rewrites one side without changing its value. Adding or subtracting the same expression on both sides keeps the balance. Dividing both sides by 7 is safe because 7 is not zero.

Work without copying the example. When finished, use the relevant focused calculator or formula reference to check the setup and result, then correct the first line where your reasoning changed. When the method feels reliable, move to algebra practice questions.

Questions about linear equations

What does it mean if I end up with something like 4 = 9?

It means the equation has no solution. Both variable terms cancelled and left a false sentence, so no number can make the original equation true. On a graph the two sides are parallel lines that never cross. If instead you land on a true sentence like $0=0$, every real number is a solution.

Does it matter which side I collect the variables on?

No, the answer is the same either way. In $2x+12=3x-5$ you can subtract $2x$ to get $12=x-5$, or subtract $3x$ to get $-x+12=-5$. Both give $x=17$. Collecting on the side with the larger coefficient just saves you from working with a negative variable term.

Can a linear equation have two different solutions?

No. A linear equation has one solution, no solution, or infinitely many. Two straight lines cannot cross at exactly two points. If you found two answers, one of them will fail the check, which usually means a sign slipped somewhere.

Why should I not divide both sides by $x$?

Because $x$ might be zero, and dividing by zero is undefined. Dividing by the variable can also delete a correct answer. In $x^2=5x$, dividing by $x$ leaves $x=5$ and hides the solution $x=0$. Move everything to one side and factor instead.