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 \(y\):

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\(a_1x+b_1y+c_1=0\)

\(a_2x+b_2y+c_2=0\)

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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