# How to Find the Number of Solutions in a System of Equations?

Depending on how the linear equations in a system touch each other, there will be a different number of solutions to the system. Here you get familiarized with how to find the number of solutions in a system of equations.

A system of linear equations usually has a single solution, but sometimes it can have no solution (parallel lines) or infinite solutions (same line).

## A step-by-step guide to the number of solutions in a system of equations

A linear equation in two variables is an equation of the form $$ax + by + c = 0$$ where $$a, b, c ∈ R$$, $$a$$, and $$b ≠ 0$$. When we consider the system of linear equations, we can find the number of answers by comparing the coefficients of the variables of the equations.

### Three types of solutions of a system of linear equations

Consider the pair of linear equations in two variables $$x$$ and $$y$$:

$$a_1x+b_1y+c_1=0$$

$$a_2x+b_2y+c_2=0$$

Here $$a_1$$, $$b_1$$, $$c_1$$, $$a_2$$, $$b_2$$, $$c_2$$ are all real numbers.

Note that, $$a_1^2 + b_1^2 ≠ 0, a_2^2 + b_2^2 ≠ 0$$.

• If $$\frac{a_1}{a_2}≠ \frac{b_1}{b_2}$$, then there will be a unique solution. If we plot the graph, the lines will intersect. This type of equation is called a consistent pair of linear equations.
• If $$\frac{a_1}{a_2}= \frac{b_1}{b_2}=\frac{c_1}{c_2}$$, then there will be infinitely many solutions. The lines will coincide. This type of equation is called a dependent pair of linear equations in two variables.
• If $$\frac{a_1}{a_2}= \frac{b_1}{b_2}≠\frac{c_1}{c_2}$$, then there will be no solution. If we plot the graph, the lines will be parallel. This type of equation is called an inconsistent pair of linear equations.

### The Number of Solutions in a System of Equations – Example 1:

How many solutions does the following system have?

$$y=-2x-4$$, $$y=3x+3$$

Solution:

First, rewrite the equation to the general form:

$$-2x-y-4=0$$

$$3x-y+3=0$$

Now, compare the coefficients:

$$\frac{a_1}{a_2}$$$$=-\frac{2}{3}$$

$$\frac{b_1}{b_2}$$$$=-\frac{1}{1}=1$$

$$\frac{a_1}{a_2}≠ \frac{b_1}{b_2}$$, Hence, this system of equations will have only one solution.

## Exercises for the Number of Solutions in a System of Equations

### Find the number of solutions in each system of equations.

1. $$\color{blue}{2x\:+\:3y\:-\:11\:=\:0,\:3x\:+\:2y\:-\:9\:=\:0}$$
2. $$\color{blue}{y=\frac{10}{3}x+\frac{9}{7},\:y=\frac{1}{8}x-\frac{3}{4}}$$
3. $$\color{blue}{y=\frac{8}{5}x+2,\:y=\frac{8}{5}x+\frac{5}{2}}$$
4. $$\color{blue}{y=-x+\frac{4}{7},\:y=-x+\frac{4}{7}}$$
1. $$\color{blue}{one\:solution}$$
2. $$\color{blue}{one\:solution}$$
3. $$\color{blue}{no\:solution}$$
4. $$\color{blue}{infinitely\:many\:solutions}$$

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