What makes an equation "linear"?

The variable appears only to the first power — no x^2, no \sqrt{x}, no \frac{1}{x}. Its graph is always a straight line, which is where the name comes from.

What if the variable cancels out completely?

Read what remains. A true statement like 5 = 5 means infinitely many solutions (the two sides are the same line). A false statement like 5 = -2 means no solution (parallel lines that never meet).

Which side should I collect the variables on?

Either works, but collecting them on the side with the larger coefficient keeps the coefficient positive and avoids sign errors — from 5x - 2 = 2x + 13, subtract 2x rather than 5x.

How are linear equations different from two-step equations?

A two-step equation like 4x - 5 = 19 is the simplest linear equation. The general case adds parentheses, fractions, and variables on both sides — more simplifying before the same final two steps.

Can Photo Math Solver handle fractions and parentheses?

Yes — it reads the equation exactly as printed, distributes and clears denominators explicitly, and labels each transformation so you can see why every line follows from the last.

You can start solving linear equations for free by adding the Chrome extension — try it on your own homework before deciding if you need anything more.

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Photo Math Solver Solved

2x - 9 = 5

1Add $9$ to both sides: $2x = 14$ .
2Divide both sides by $2$ : $x = 7$ .
3Check: $2(7) - 9 = 5$ ✓.

On this page01Definition 02Formulas 03Methods 04Examples 05Practice 06Mistakes 07FAQ

01Definition

What is a linear equation?

A linear equation is an equation where the variable appears only to the first power — no $x^2$ , no $\sqrt{x}$ , no $\frac{1}{x}$ . Graphed, every linear equation draws a straight line, which is where the name comes from.

The general shape you will meet is $ax + b = cx + d$ : variable terms and constants can sit on both sides, sometimes wrapped in parentheses or fractions. Solving means legally rearranging — doing the same thing to both sides — until the variable stands alone.

Unlike one-step and two-step equations, a full linear equation usually needs simplifying first: distribute parentheses, combine like terms on each side, and only then start moving terms across the equals sign.

Where you'll actually use this

Budgeting: "a $30 base fee plus $12 per month equals $126 — how many months?" is a linear equation.
Unit conversion and pricing: comparing two phone plans or ride fares means solving where two linear expressions are equal.
Physics: constant-speed motion problems ( $d = vt + d_0$ ) are linear in time.
The slope-intercept form $y = mx + b$ underpins trend lines and simple predictions in statistics.

02Formulas

The formulas you need

Keep these on hand — every method below builds on them.

General form

ax + b = cx + d

Variables and constants on both sides — collect variables on one side, constants on the other.

Solution of ax + b = c

x = \frac{c - b}{a}

Subtract b from both sides, then divide by a — every linear solve reduces to this.

Slope-intercept form

y = mx + b

The two-variable cousin: m is the slope, b is where the line crosses the y-axis.

Special cases

0 = 0 \;\Rightarrow\; \infty \text{ solutions}, \quad 0 = k \;\Rightarrow\; \text{none}

If the variable cancels entirely, the leftover statement decides: true → infinitely many, false → no solution.

03Methods

How to solve linear equations

Pick the method that fits the problem in front of you.

When to use it: When all variable terms already sit on one side, like

3x + 7 = 22

— the simplest case and the pattern every other case reduces to.

1Undo addition or subtraction first: move the constant to the other side.
2Undo multiplication or division: divide both sides by the coefficient.
3Check by substituting the answer back into the original equation.

04Worked examples

Linear Equations, solved step by step

From easy to hard — pick a problem to see its full solution.

2x - 9 = 5

1Add $9$ to both sides: $2x = 14$ .
2Divide both sides by $2$ : $x = 7$ .
3Check: $2(7) - 9 = 5$ ✓.

Try it on your own linear equations homework

Photo Math Solver reads a linear equation straight from a screenshot — a worksheet PDF, a textbook page, an online quiz — and returns the full solution: distributing, combining like terms, and isolating the variable, one labeled step at a time. No losing a sign while retyping, and no wondering which term to move first.

05Practice

Now you try

Work each one on paper first, then check your answer.

Problem 1

7x - 4 = 24

Problem 2

3(x - 2) = x + 8

Problem 3

6x + 5 = 2x - 11

Problem 4

\frac{x}{5} - 1 = 2

Stuck on your linear equations homework? Screenshot it

When a linear equation has variables on both sides plus parentheses to distribute, it is easy to lose track of a sign while retyping it. Screenshot it with Photo Math Solver instead — it captures the equation exactly as written and walks through combining and isolating terms the same way.

06Common mistakes

Where points get lost

Each of these shows up on real graded work — and each has a simple fix.

The mistake	Why it happens	The fix
Moving a term across the equals sign without flipping its sign.	"Move it over" is taught as a shortcut, and the sign flip that justifies it gets forgotten under time pressure.	Think "subtract from both sides," not "move." Writing the operation on both sides makes the sign change automatic.
Distributing into the first term only: $2(x + 3)$ becoming $2x + 3$ .	The eye latches onto the first term inside the parentheses and moves on.	Draw the two distribution arrows before writing anything: $2 \cdot x$ and $2 \cdot 3$ , giving $2x + 6$ .
Losing a negative when distributing a minus: $-2(x - 4)$ becoming $-2x - 8$ .	Two sign rules stack ( $-2$ times $-4$ ), and one of them gets dropped.	Multiply signs explicitly: $(-2)(-4) = +8$ , so $-2(x - 4) = -2x + 8$ .
Dividing only one term by the coefficient at the last step.	After several steps the equation looks solved, and the division gets applied to just the nearest term.	The final division applies to the entire side. From $3x = 15 + 6$ , first combine ( $3x = 21$ ), then divide everything: $x = 7$ .

Photo Math Solver shows every intermediate step, so slips like these are easy to catch before they cost you marks.

07FAQ