Linear Equations

A linear equation expresses a constant-rate relationship. In one variable it asks for the value that makes a statement true, often in the form $ax + b = c$ with $a \ne 0$. In two variables it describes a straight line on a graph.

The formula and what the symbols mean

The one-variable template is

where $a$ is the coefficient of the variable, $b$ a constant, and $c$ the target value. Solving it isolates $x$. The two-variable form, such as $y = 2x + 5$, instead links $x$ and $y$ so that its graph is a line.

An equation is linear when every variable term is first degree: no squares like $x^2$, no products like $xy$ in introductory algebra, and no variables in denominators. So these are linear:

• $3x - 7 = 11$
• $y = 2x + 5$
• $4x + 3y = 12$

and these are not:

• $x^2 + 1 = 0$
• $xy = 10$
• $y = \frac{1}{x}$

Why a linear equation graphs as a straight line

The defining feature is constant change. If equal changes in $x$ always produce equal changes in $y$, the relationship is linear. For $y = 2x + 1$, every time $x$ rises by $1$, $y$ rises by $2$ — a fixed rate of change traces a straight line rather than a curve.

When the rate of change is not constant, the graph bends. For example, $y = x^2$ curves upward because the change in $y$ grows as $x$ increases. This is exactly why hourly pay, steady speed, and "starting value plus the same change each step" all model as lines.

Worked example: solve $3x - 7 = 11$

Undo the operations around $x$ in reverse order. Add $7$ to both sides:

Divide both sides by $3$:

Check in the original equation:

The check holds, so $x = 6$. That is the core move: isolate the variable, then verify in the original.

Practice with a check

Solve $5x + 4 = 19$ and confirm by substitution — you should find $x = 3$, since $5(3) + 4 = 19$. For a second case, rewrite $y - 3 = 2x$ as $y = 2x + 3$ and describe what happens to $y$ when $x$ increases by $1$. If $y$ climbs by exactly $2$ each step, you have read the constant rate correctly.

Calculation pitfalls

One pitfall is treating any equation with both $x$ and $y$ as linear. That holds only when the variables stay first degree and the relationship has constant change.

Another is operating on just one side. If you add, subtract, multiply, or divide on the left, you must do the same on the right to keep the equation balanced.

A third is dividing by a coefficient without checking the condition. In $ax + b = c$, solving by division assumes $a \ne 0$; if $a = 0$ it is no longer a standard one-variable linear equation. Linear equations appear wherever a quantity changes at a steady rate — budgeting, distance-and-time problems, unit pricing, simple physics — and they are often the first useful model because they are easy to solve, graph, and interpret.