First-Degree Equations

A first-degree equation is an equation where the unknown appears only to the first power. The most common form is

$ax + b = 0$

with the condition $a \ne 0$. Here $a$ is the coefficient that multiplies the unknown $x$, and $b$ is the constant term. The goal is to find the value of $x$ that makes the equality true.

The key point isn't just having a letter. The key point is that the unknown appears with an exponent of $1$. For example, $3x + 4 = 19$, $7 - 2x = 1$, and $\frac{x}{5} + 6 = 10$ are all first-degree equations. On the other hand, $x^2 + 3 = 12$ is not first-degree because the unknown is squared.

Why solving means keeping the balance

Think of the equation as a balanced scale. If you subtract $5$ from one side, you must subtract $5$ from the other. If you divide one side by $3$, you must divide the other by $3$ as well.

This is why isolating $x$ works. Solving the equation isn't about "moving terms" magically; it's about applying equivalent operations to both sides until the unknown is left alone. From this idea you can also read off the general result directly: starting from

$ax + b = 0$

with $a \ne 0$, subtract $b$ from both sides to get

$ax = -b$

and divide both sides by $a$ to get

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

This formula only works when the equation is truly first-degree, meaning when $a \ne 0$.

Worked example: solving step by step

Solve:

$3x + 5 = 17$

First, subtract $5$ from both sides:

$3x = 12$

Now divide both sides by $3$:

$x = 4$

To check your answer, substitute $x = 4$ back into the original equation:

$3(4) + 5 = 12 + 5 = 17$

Since the equality is true, the solution is correct.

Try these and check your answers

Solve each one, then substitute your result back into the original equation to confirm it.

1. $5x - 8 = 22$. Subtract and divide as above; the check should give $22$.
2. $7 - 2x = 1$. Watch the sign on the $x$ term as you isolate $x$.

Working the substitution check every time is the habit that catches most mistakes before they cost you.

Calculation traps to avoid

A common trap is swapping a sign without thinking about the operation being performed. Instead of memorizing that a number "moves to the other side by changing its sign," it is safer to say: I will add or subtract the same value from both sides.

Another error is forgetting to divide all terms correctly. If $3x = 12$, then $x = 4$, not $x = 9$ or $x = 12 - 3$.

It's also important to watch the condition $a \ne 0$. If in $ax + b = 0$ we have $a = 0$, the equation is no longer first-degree, and the type of problem changes.

Where this appears

First-degree equations appear in simple problems involving price, age, distance, unit conversion, and comparisons between quantities. Whenever there is a linear relationship between an unknown and known numbers, this model usually pops up. They are also the foundation for many later algebra topics because they train the concepts of equivalence and variable isolation.