Linear Equations

A linear equation shows 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 key idea is constant change. If equal changes in $x$ always produce equal changes in $y$, the relationship is linear. That is why linear equations show up in hourly pay, steady speed, and any situation with a starting value plus the same change each step.

What Makes a Linear Equation Linear

An equation is linear if every variable term is first degree. That means no squares like $x^2$, no products like $xy$ in an introductory algebra setting, and no variables in denominators.

For example, these are linear:

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

These are not linear:

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

Two Common Cases: Solving and Graphing

In a one-variable problem, the goal is usually to solve for the unknown number. In $3x - 7 = 11$, you want the value of $x$ that makes the equation true.

In a two-variable problem, the goal is often to understand a relationship. In $y = 2x + 5$, the equation tells you how $y$ changes when $x$ changes, and its graph is a line.

Worked Example: Solve $3x - 7 = 11$

Start by undoing the operations around $x$ in reverse order.

Add $7$ to both sides:

$$3x - 7 + 7 = 11 + 7$$

so

$$3x = 18$$

Now divide both sides by $3$:

$$x = 6$$

Check the answer in the original equation:

$$3(6) - 7 = 18 - 7 = 11$$

The check works, so $x = 6$ is correct. This is the core move in solving a linear equation: isolate the variable, then verify the result in the original equation.

Why the Graph Is a Straight Line

For a two-variable linear equation such as $y = 2x + 1$, the change in $y$ stays constant. If $x$ goes up by $1$, then $y$ goes up by $2$ every time. A constant rate of change gives you a straight line instead of a curve.

If the rate of change is not constant, the graph will usually not be a line. For example, $y = x^2$ curves upward because the change in $y$ gets larger as $x$ increases.

Common Mistakes When Solving Linear Equations

One common mistake is treating any equation with both $x$ and $y$ as linear. That only works if the variables stay to the first power and the relationship has constant change.

Another mistake is doing an operation on only one side when solving. If you add, subtract, multiply, or divide on the left side, you must do the same thing on the right side to keep the equation balanced.

A third mistake is dividing by a coefficient without checking the condition. In $ax + b = c$, solving by division assumes $a \ne 0$. If $a = 0$, the equation is no longer a standard linear equation in one variable.

Where Linear Equations Are Used

Linear equations appear whenever a quantity changes at a steady rate. You see them in budgeting, distance-and-time problems, unit pricing, and simple physics models.

They are often the first useful model because they are simple to solve, simple to graph, and easy to interpret. Over a limited range, even more complicated data is often approximated with a line.

Try a Similar Problem

Try solving $5x + 4 = 19$ and check your answer by substitution. If you want to explore another case, rewrite $y - 3 = 2x$ as $y = 2x + 3$ and describe what happens to $y$ when $x$ increases by $1$.