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).

Related Topics

• How to Solve Systems of Equations
• How to Solve Systems of Equations Word Problems

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.

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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. \(\color{blue}{infinitely\:many\:solutions}\)