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

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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?

• \(\color{blue}{11x+12=17x}\)

• \(\color{blue}{no\:solution}\)