Simultaneous Linear Equations
Simultaneous linear equations in two variables involve two unknown quantities to represent reallife problems.
It helps in establishing a relationship between quantities, prices, speed, time, distance, etc results in a better understanding of the problems.
We all use simultaneous linear equations in our daily life without knowing it.
In this minilesson, we will learn in detail about the solutions of linear equations, consistent and inconsistent equations, homogenous linear equations, simultaneous equation solver, example of simultaneous equation, etc. in this minilesson.
Lesson Plan
What Do You Mean By Simultaneous Linear Equation?
Two linear equations in two or three variables solved together to find a common solution are called simultaneous linear equations.
For example, We can visualize the solution of the linear system of equations by drawing 2 linear graphs and finding out their intersection point.
The red dot represents the solutions for equation 1, and equation 2. The intersection is the unique point (2,1) is the solution that we are looking for which will satisfy both the equations
Linear Equation Calculator
This simultaneous equation solver will find the nontrivial solution on entering the coefficients of x and y and constant.
How To Solve Simultaneous Linear Equations?
The following methods can be used to find the solution of linear system of equations, let's see some example of the simultaneous equation.
1. Substitution Method
Consider the following pair of linear equations:
\[\begin{array}{l}x + 2y = 6\;\;\;\; & ...(1)\\ x  y = 3\;\;\;\; & ...(2)\end{array}\]Let’s rearrange the first equation to express \(x\) in terms of \(y\), as follows:\[\begin{array}{l}x + 2y = 6\\ \Rightarrow \;\;\;x = 6  2y\end{array}\]
This expression for \(x\) can now be substituted in the second equation, so that we will be left with an equation in \(y\) alone:
\[\begin{align}& x  y = 3\\ &\Rightarrow \;\;\;6  2y y = 3\\ &\Rightarrow \;\;\;3y = 3  6\\ &\Rightarrow \;\;\;y = \frac{3}{3}\\ &\Rightarrow \;\;\;y = 1\end{align}\]
Once we have the value of \(y\), we can plug this back into any of the two equations to find out \(x\). Lets plug it into the first equation:
\[\begin{array}{l}x + 2y = 6\\ \Rightarrow \;\;\;x + 2 \times 1 = 6\\ \Rightarrow \;\;\;x = 6  2 = 4\\ \Rightarrow \;\;\;x = 4\end{array}\]
The final nontrivial solution is:
\[x = 4,\;y = 1\]
It should be clear why this process is called substitution. We express one variable in terms of another using one of the pair of equations, and substitute that expression into the second equation.
2. Elimination Method
Consider the following pair of linear equations:
\[\begin{array}{l} 2x + 3y  7 = 0\\ 3x + 2y  3 = 0 \end{array}\]
The coefficients of x in the two equations are 2 and 3 respectively. Let us multiply the first equation by 3 and the second equation by 2, so that the coefficients of x in the two equations become equal:
\[\begin{array}{l}\left\{ \begin{array}{l}3 \times \left( {2x + 3y  7 = 0} \right)\\2 \times \left( {3x + 2y  3 = 0} \right)\end{array} \right.\\ \Rightarrow \;\;\;6x + 9y  21 = 0\\\qquad6x + 4y  6 = 0\end{array}\]
Now, let us subtract the two equations, which means that we subtract the left hand sides of the two equations, and the right hand side of the two equations, and the equality will still be preserved (this should be obvious: if I = II and III = IV, then I – III will be equal to II – IV):
\[\begin{array}{l}\left\{ \begin{array}{l}\,\,\,\,6x + 9y  21 = 0\\\,\,\,\,6x + 4y  6 = 0\\\underline {  \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,} \\\,\,\,\,\,\,0 + 5y\,\,\,  15 = 0\end{array} \right.\\ \Rightarrow \;\;\;5y = 15\\ \Rightarrow \;\;\;y = 3\end{array}\]
Note how x gets eliminated, and we are left with an equation in y alone. Once we have the value of y, we proceed as earlier – we plug this into any of the two equations. Let us put this into the first equation:
\[\begin{array}{l}2x + 3y  7 = 0\\ \Rightarrow \;\;\;2x + 3\left( 3 \right)  7 = 0\\ \Rightarrow \;\;\;2x + 9  7 = 0\\ \Rightarrow \;\;\;2x = 2\\ \Rightarrow \;\;\;x = 1\end{array}\]
Thus, the solution is:
\[x = 1,\;y = 3\]
3. Graphical Method
As an example, consider the following pair of linear equations:
\[\begin{array}{l} x  y = 0\\ x + y  4 = 0\end{array}\]
We draw the corresponding lines on the same axes:
The point of intersection is \(A\left( {2,\,\,2} \right)\), which means that \(x = 2,\;\;y = 2\) is a solution to the pair of linear equations given by (2). In fact, it is the only solution to the pair, as two nonparallel lines cannot intersect in more than one point.

You can check directly about the types of solution using the following conditions:
Unique solution( Consistent and independent) \[\frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{2}}\]
No solution( Inconsistent and independent) \[\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}}\]
Infinite Many Solutions (Consistent and Dependent) \[\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}\]
Solved Examples
Example 1 
The sum weights of Fabia and Valerian is 60 pounds and the difference is 2. Find the weights of Fabia and Valerian.
Solution
Let the weight of Fabia and Valerian be \(x\) pounds and \(y\) pounds respectively.
Therefore, the simultaneous equations are\[\begin{array}{l}x + y = 60\;\;\;\; & ...(1)\\ x  y = 2\;\;\;\; & ...(2)\end{array}\]
Let’s rearrange the first equation to express \(x\) in terms of \(y\), as follows:\[\begin{array}{l}x + y = 60\\ \Rightarrow \;\;\;x = 60  y\end{array}\]
This expression for \(x\) can now be substituted in the second equation, so that we will be left with an equation in \(y\) alone:
\[\begin{align}& x  y = 2\\ &\Rightarrow \;\;\;60  y y = 2\\ &\Rightarrow \;\;\;2y = 2  60\\ &\Rightarrow \;\;\;y = \frac{58}{2}\\ &\Rightarrow \;\;\;y = 29\end{align}\]
Once we have the value of \(y\), we can plug this back into any of the two equations to find out \(x\). Lets plug it into the first equation:
\[\begin{array}{l}x + y = 60\\ \Rightarrow \;\;\;x + 29 = 60\\ \Rightarrow \;\;\;x = 60  29\\ \Rightarrow \;\;\;x = 31\end{array}\]
The final solution is:
\[x = 31,\;y = 29\]
Therefore, weight of Fabia and Valerian is 31 pounds and 29 pounds respectively. 
Example 2 
Can you help Alex to find a twodigit number whose units digit is thrice the tens digit and if 36 is added to the number, the digits interchange their place.
Solution
Let the digit in the units place is \(x\).
And the digit in the tens place be \(y\)
Then \(x = 3y\) and the number \(= 10y + x\)
The number obtained by reversing the digits is \(10x + y\).
If \(36\) is added to the number, digits interchange their places,
Therefore, we have \[10y + x + 36 = 10x + y\]
\[10y – y + x + 36 = 10x + y  y\]
\[9y + x – 10x + 36 = 10x  10x\]
\[9y  9x + 36 = 0 \]
\[9(x  y) = 36\]
\[ x  y = \frac{36}{9}\]
\[x  y += 4 .... (i) \]
Substituting the value of x = 3y in equation (i), we get
\[3y  y = 4\]
\[ 2y = 4\]
\[ y = \frac{4}{2}\]
\[ y = 2\]
Substituting the value of\(y = 2\)in equation (i),we get
\[ x  2 = 4 \]
\[ x = 4 + 2 \]
\[ x = 6\]
Therefore, the number Alex looking is 62. 
Solved Crossword Puzzle on Simultaneous linear equations
Interactive Questions
Here are a few activities for you to practice. Select/Type your answer and click the "Check Answer" button to see the result.
Let's Summarize
The minilesson targeted the fascinating concept of Simultaneous Linear Equations. The math journey around Simultaneous Equations starts with what a student already knows, and goes on to creatively crafting a fresh concept in the young minds. Done in a way that not only it is relatable and easy to grasp, but also will stay with them forever. Here lies the magic with Cuemath.
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FAQs on Simultaneous linear equations
1. What is the degree of a linear equation?
The degree of a linear equation is 1
2. How many types of linear equations are there?
The three major forms of linear equations are pointslope form, standard form, and intercept form.
3. What are linear equations?
A linear equation is an equation in which the variable(s) is(are) with the exponent 1
Example: \[2 x = 23 \]
\[ x  y = 5\]
4. How does one solve the system of linear equations?
We have different methods to solve the system of linear equations:
Graphical Method
Substitution Method
Cross Multiplication Method
Elimination Method
Determinants Method
5. What is the formula of linear equations in two variables?
The general equation of linear equation in two variables is \[ ax + by + c = 0 \]
6. Can a linear equation have 2 solutions?
No, the system of the linear equation may have unique or one, no or zero, and an infinite number of solutions.