How to Solve Linear Equations in Two Variables?

A linear equation in two variables is a linear equation in which there are two variables. In the following guide, you will learn more about linear equations in two variables and solving them.

How to Solve Linear Equations in Two Variables?

We can recognize a linear equation in two variables that are expressed as \(ax+by+c=0\), which consists of two variables \(x\) and \(y\) and the highest degree of the given equation is \(1\).

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Step by step guide to linear equations in two variables

The standard form of a two-variable linear equation is \(ax+ by+ c= 0\) where \(x\) and \(y\) are the two variables. Graphical representation of linear equations in two variables consists of two straight lines that can be intersecting lines, parallel lines, or coincident lines.

Forms of linear equations in two variables

A linear equation in two variables can be in different forms such as standard form, intercept form, and point-slope form. Look at the image below, which shows all three forms of representation of linear equations in two variables with examples.

Methods for solving linear equations in two variables

There are five methods to solve a system of linear equations in two variables:

  • Graphical Method
  • Substitution Method
  • Cross Multiplication Method
  • Elimination Method

Graphical method

The steps for solving linear equations in two variables graphically are presented below:

  • Step 1: To solve a system of two equations in two variables graphically, we graph each equation.
  • Step 2: To graph an equation manually, first convert it to the form \(y=mx+b\) by solving the equation for \(y\).
  • Step 3: Start putting the values of \(x\) as \(0, 1, 2,\) and so on and find the corresponding values of \(y\), or vice-versa.
  • Step 4: Identify the point where both lines meet.
  • Step 5: The point of intersection is the solution of the given system.

Example: Find the solution of the following system of equations graphically.
\(-x+2y-3 =0\)
\(3x+4y-11=0\)
Solution: 

We plot them and see if they intersect at one point. As you can see below, both lines meet in \((1, 2)\). Therefore, the answer to the given linear equation system is \(x=1\) and \(y=2\).

But the two lines may not always intersect. Sometimes they may be parallel. In that case, the system of linear equations in the two variables has no answer. In some other cases, both lines coincide with each other. In that case, each point on that line is a solution to the given system and hence the given system has an infinite number of solutions. If the system has a solution, then it is said to be consistent; otherwise, it is said to be inconsistent.

Consider a system of two linear equations: \(a_1x + b_1y+ c_1= 0\) and \(a_2x + b_2y+ c_2=0\):

Method of substitution

To solve a system of two linear equations in two variables using the substitution method, we must use the following steps:

  • Step 1: Solve one of the equations for one variable.
  • Step 2: Substitute this in another equation to get an equation based on a single variable.
  • Step 3: Solve it for the variable.
  • Step 4: Substitute it in any of the equations to get the value of another variable.

Cross multiplication method

Consider a system of linear equations: \(a_1x + b_1y+ c_1= 0\) and \(a_2x + b_2y+ c_2=0\)

To solve this problem using the cross-multiplication method, we first write the coefficients of each of \(x\) and \(y\) and the constants as follows:

Here, the arrows indicate that those coefficients must be multiplied. Now we write the following equation by multiplication and subtraction.

\(\frac{x}{b_1 c_2- b_2 c_1}=\frac{y}{c_1 a_2 – c_2 a_1}=\frac{1}{a_1 b_2 – a_2 b_1}\)

From this equation, we get two equations:

\(\frac{x}{b_1 c_2- b_2 c_1}=\frac{1}{a_1 b_2 – a_2 b_1}\)

\(\frac{y}{c_1 a_2 – c_2 a_1}=\frac{1}{a_1 b_2 – a_2 b_1}\)

Solving each of these for \(x\) and \(y\), the solution of the given system is:

\(x=\frac{b_1c_2−b_2c_1}{a_1b_2−a_2b_1}\)

\(y=\frac{c_1a_2−c_2a_1}{a_1b_2−a_2b_1}\)

Method of elimination

To solve a system of linear equations in two variables using the elimination method, we use the following steps:

  • Step 1: Arrange the equations in the standard form: \(ax+by+c=0\) or \(ax+by=c\).
  • Step 2: Check if adding or subtracting the equations would result in the cancellation of a variable.
  • Step 3: If not, multiply one or both equations by either the coefficient of \(x\) or \(y\) so that their addition or subtraction would result in the cancellation of any one of the variables.
  • Step 4: Solve the resulting single variable equation.
  • Step 5: Substitute it in any of the equations to get the value of another variable.

Note:

When solving equations using the substitution method or the elimination method:

  • If we get an equation that is true (i.e., something like \(0 = 0\), \(-1 = -1\), etc), then it means that the system has an infinite number of solutions.
  • If we get an equation that is false (i.e., something like \(0 = 2\), \(3= -1\), etc), then it means that the system has no solution.

Linear Equations in Two Variables – Example 1:

Solve the following system of equations using the substitution method.

\(x+2y-7=0\)
\(2x-5y+13=0\)

Solution: 

First, solve the equation, \(x+2y-7=0\) for \(y\):
\(x+2y-7=0\)
\(2y=7-x\)
\(y=\frac{(7-x)}{2}\)

Now, substitute this in the equation, \(2x-5y+13=0\):
\(2x-5y+13=0\)
\(2x-5(\frac{(7-x)}{2})+13=0\)
\(2x-(\frac{35}{2})+(\frac{5x}{2})+13=0\)
\(2x + (\frac{5x}{2}) = \frac{35}{2} – 13\)
\(\frac{9x}{2} = \frac{9}{2}\)
\(x=1\)

Substitute \(x=1\) this in the equation \(y=\frac{(7-x)}{2}\):
\(y=\frac{(7-1)}{2} = 3\)
Therefore, the solution of the given system is \(x=1\) and \(y=3\).

Exercises for Linear Equations in Two Variables

  1. A boat running downstream covers a distance of \(20 km\) in \(2\) hours while covering the same distance upstream, takes \(5\) hours. What is the speed of the boat in still water?
  2. Solve the system of equations using the elimination method. \(2x+3y-9=0,\:3x+2y-11=0\)
  3. Find the solution of the following system of equations graphically. \(x−y=0,\:x+y−6=0\)
This image has an empty alt attribute; its file name is answers.png
  1. \(\color{blue}{7 \frac{km}{h}}\)
  2. \(\color{blue}{x=3 , y=1}\)
  3. \(\color{blue}{x=3 , y=3}\)

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