Table of Contents
Linear equations are one of the first truly “algebraic” objects you work with. In this chapter, we focus on what a linear equation in one variable looks like and how to solve it in a clear, step‑by‑step way.
Because “Variables and Expressions” has its own chapter, we will assume you are already comfortable with what a variable is and what an algebraic expression is. Here we focus on equations and, in particular, linear ones in preparation for the two subchapters: “One-variable equations” and “Word problems.”
What is a linear equation (in one variable)?
An equation is a statement that two expressions are equal, usually involving a variable, like
$$
3x + 5 = 11.
$$
A linear equation in one variable is an equation where:
- there is only one variable (often $x$),
- the highest power of the variable is $1$ (no $x^2$, $x^3$, etc.),
- the variable is not in a denominator or under a root.
The general form can be written as
$$
ax + b = c
$$
or, more generally,
$$
ax + b = cx + d,
$$
where $a$, $b$, $c$, and $d$ are numbers (constants), and $x$ is the variable.
Some examples of linear equations in one variable:
- $2x + 3 = 11$
- $5 - 4x = 1$
- $7x - 2 = 3x + 10$
- $-3x = 9$
Non-examples (these are not linear):
- $x^2 + 2 = 0$ (variable squared)
- $\dfrac{1}{x} + 3 = 5$ (variable in denominator)
- $\sqrt{x} + 1 = 4$ (variable inside a square root)
In this chapter, we work only with linear equations in a single variable.
The goal of solving a linear equation
To solve a linear equation means to find all values of the variable that make the equation true.
For example, in
$$
3x + 5 = 11,
$$
we want to find the number $x$ such that “$3x + 5$ is equal to $11$” is a true statement.
The solution is the value of $x$ that works. If $x = 2$, then
$$
3(2) + 5 = 6 + 5 = 11,
$$
so $x = 2$ is a solution.
The solution set is the set of all solutions. For a simple linear equation in one variable, the solution set typically contains:
- exactly one number (a unique solution),
- no number at all (no solution),
- or every real number (infinitely many solutions).
Recognizing which case you have is part of solving linear equations.
Basic idea: keeping the equation balanced
An equation is like a balance scale: both sides must be equal (balanced). Any move you make to both sides of the equation preserves the balance.
Key idea:
If you perform the same operation on both sides of an equation, the equation remains true.
The main types of operations we use for solving linear equations are:
- Adding or subtracting the same number or expression on both sides.
- Multiplying or dividing both sides by the same nonzero number.
These operations are used to “undo” or isolate parts of the equation until the variable stands alone on one side.
General strategy to solve a linear equation
While the next subchapters will go into detail and examples, it is helpful to have an overall picture of the usual steps. For a typical linear equation in one variable:
- Simplify each side separately
- Use arithmetic with numbers.
- Combine like terms on each side.
- Use the distributive property (if needed) to remove parentheses.
- Get all variable terms on one side
- Add or subtract terms with the variable from both sides to gather them on one side.
- Get all constant terms on the other side
- Add or subtract numbers (constants) from both sides to gather them on the other side.
- Isolate the variable
- Multiply or divide both sides by a number to get $x$ alone (like $x = \text{number}$).
- Check your solution
- Substitute the found value of $x$ back into the original equation to verify both sides are equal.
The essential idea is: step by step, undo the operations happening to $x$ until it is alone.
What kinds of solutions can a linear equation have?
Even though the equations you meet at first usually have exactly one solution, more generally a linear equation in one variable can lead to three situations:
- One solution
You end up with something like
$$
x = 4
$$
or
$$
x = -\dfrac{1}{2}.
$$
In this case, the solution set has exactly one number. - No solution
In the process of solving, all $x$ terms cancel out, and you get a false statement, like
$$
0 = 5
$$
or
$$
3 = -1.
$$
Since this can never be true, there is no value of $x$ that makes the original equation true. The solution set is empty. - Infinitely many solutions
All $x$ terms cancel, and you get a true statement, like
$$
0 = 0
$$
or
$$
5 = 5.
$$
This means every real number $x$ makes the original equation true. The solution set is “all real numbers.”
Recognizing which result you have is part of understanding linear equations, and it prepares you for more complex equations later.
Common features of linear equations you will see
In the subchapters, you will work with many standard forms. Here is what you might encounter, so you can recognize them quickly:
- Simple one-step equations (solve in a single operation), like:
- $x + 5 = 9$
- $3x = 15$
- $x - 4 = -2$
- $\dfrac{x}{6} = 3$
- Two-step equations, like:
- $2x + 3 = 11$
- $5 - 3x = 8$
- $\dfrac{x}{4} - 2 = 1$
- Equations with variables on both sides, like:
- $3x + 5 = x + 11$
- $7 - 2x = 4x + 1$
- Equations with parentheses (requiring distribution), like:
- $2(x + 3) = 14$
- $5(2x - 1) = 3x + 7$
- Equations with fractions (to be simplified), like:
- $\dfrac{x}{3} + \dfrac{1}{2} = \dfrac{5}{6}$
- $\dfrac{2x - 1}{4} = 3$
All of these are still linear because:
- there is only one variable,
- the variable is to the first power,
- the variable is not inside a denominator with another variable (in this level, if fractions appear, numerators and denominators are just numbers or simple expressions that keep the equation linear).
Connections to word problems and real-life situations
Many real-world situations turn into linear equations when you translate the words into algebra. For example:
- “A taxi charges a base fee plus a cost per kilometer” leads to a linear relationship between distance and cost.
- “You save a fixed amount of money every week” leads to a linear relationship between weeks and total savings.
The chapter “Word problems” under “Linear Equations” will focus on how to:
- turn a story or situation into a linear equation, and
- solve that equation to answer the question asked.
For now, the important point is: a lot of everyday situations that grow or change at a steady rate can be modeled by linear equations.
Why linear equations matter
Linear equations are foundational because they:
- give you a systematic way to find unknown quantities,
- appear in almost every later area of mathematics (including functions, graphing lines, systems of equations),
- make it possible to model and solve many simple real-world problems.
They are the main bridge between arithmetic (working only with numbers) and full algebra (working with general symbols and relationships).
In the next subchapters, you will:
- practice solving many types of one-variable linear equations, and
- learn how to set up and solve word problems that lead to such equations.