# How to Find the Number of Solutions to a Linear Equation?

A linear equation is an equation in which the largest power of the variable is equal to \(1\). In this article, in a few short steps, we will explain the method of identifying the number of solutions of a linear equation.

**Related Topics**

- How to Convert a Linear Equation in Standard Form to Slope-Intercept Form?
- How to Write the Standard Form of Linear Equations?

**A step-by-step guide to Finding the Number of Solutions to a Linear Equation**

An algebraic expression in which there is equality is called an equation. An equation in which there is only one variable is called a single unknown equation. If the highest variable exponent in the equation is equal to \(1\), we call this equation a linear equation.

To solve a linear equation, we first move the variables to one side and the known numbers to the other side of the equation. After that, we simplify the expression. According to the statement obtained after simplification, there are \(3\) different modes for solving the linear equation:

If the statement obtained is false, there is no solution to the linear equation.

If the obtained statement is true for only one value, then the linear equation has a solution.

If the obtained statement is always true, then there are infinite solutions for the linear equation.

**Finding the Number of Solutions to a Linear Equation-Example 1**

How many solutions does the following equation have?

\(-12x+25=-12x\)

**Solution:**

Solve for \(x: -12x+25=-12x\)→\(25=+12x-12x\)→\(25=0\).

\(25=0\) is a false statement. The linear equation has no solution because by solving the linear equation you get a false statement as an answer.

**Finding the Number of Solutions to a Linear Equation-Example 2**

How many solutions does the following equation have?

\(10x=32x-22x\)

**Solution:**

Simplify the right side of the equation: \(10x=32x-22x\)→\(10x=10x\).

\(10x=10x\) is a statement that is always true. So, the equation has infinitely many solutions because by solving a linear equation you get a statement that is always true.

**Exercises for** **Finding the Number of Solutions to a Linear Equation**

**How many solutions does the following equation have?**

- \(\color{blue}{11x+12=17x}\)

- \(\color{blue}{no\:solution}\)

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