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Book Free DemoWhat are simultaneous linear equations?
A set of equations with two or more linear equations having the same variables is called as simultaneous linear equations or system of linear equations or a pair of linear equations.
\(2x+y=1\) and \(x-y=3\)
Together they are called as simultaneous linear equations.
Example:
Jane bought \(2\) apples and \(1\) banana for a total cost of \($8\). Let us frame an equation to find the individual cost of an apple and a banana.
Let us understand the purpose of simultaneous linear equations with a real life situation.
Let \(x\) denote the cost of an apple and \(y\) denote the cost of a banana.
Writing in equation, she has:
\(2x+y=8\) ---- \((1)\)
Jane tries to find the value of each apple and banana by substituting the values for \(x\).
When \(x=1\), \(2(1)+y=8\)\(\Rightarrow y=8-2\)\(\Rightarrow y=6\)
When \(x=2\), \(2(2)+y=8\)\(\Rightarrow y=8-4\)\(\Rightarrow y=4\)
When \(x=3\), \(2(3)+y=8\)\(\Rightarrow y=8-6\)\(\Rightarrow y=2\)
When \(x=4\), \(2(4)+y=8\)\(\Rightarrow y=8-8\)\(\Rightarrow y=0\)
Now, writing these values in the table, we have:
\(x\) | \(1\) | \(2\) | \(3\) | \(4\) | …. |
\(y\) | \(6\) | \(4\) | \(2\) | \(0\) | …. |
Jane plots these points in the graph and draws a line joining these points.
![Ex.10(1).gif](https://resources.cdn.yaclass.in/9c8de479-d3e3-4d90-bae0-a0c45675829b/Ex101.gif)
Thus, she gets many number of solutions. Since she is insufficient with the apples and bananas, she again went to the shop and brought \(1\) apple and \(2\) bananas for a total cost of \(₹10\).
Writing in equation, she has:
\(x+2y=10\) ---- \((2)\)
Again, she tries to find each apple and banana's value by substituting the values for \(x\).
When \(x=1\), \(1+2y=10\)\(\Rightarrow 2y=10-1=9\)\(\Rightarrow y=\frac{9}{2}=4.5\)
When \(x=2\), \(2+2y=10\)\(\Rightarrow 2y=10-2=8\)\(\Rightarrow y=\frac{8}{2}=4\)
When \(x=3\), \(3+2y=10\)\(\Rightarrow 2y=10-3=7\)\(\Rightarrow y=\frac{7}{2}=3.5\)
When \(x=4\), \(4+2y=10\)\(\Rightarrow 2y=10-4=6\)\(\Rightarrow y=\frac{6}{2}=3\)
Now, writing these values in the table, we have:
\(x\) | \(1\) | \(2\) | \(3\) | \(4\) | …. |
\(y\) | \(4.5\) | \(4\) | \(3.5\) | \(3\) | …. |
Jane plots these points in the graph and draws a line joining these points.
![Ex.10(2).gif](https://resources.cdn.yaclass.in/8d4a9b93-0544-44a8-948a-25eaa8013ec1/Ex102.gif)
In the graph, she found that the two lines intersect at the point \((2,4)\).
Hence, Jane came to a conclusion that if we solve two equations together, we get an unique solution.
By solving equations \((1)\) and \((2)\), Jane gets the cost of an apple as \($2\) and the cost of a banana as \($4\).
These two equations are called as simultaneous linear equations.
Important!
A solution to the simultaneous linear equation can be found in many ways. They are:
1. Graphical method
2. Substitution method
3. Elimination method
4. Cross multiplication method