Theory:

What 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.

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.

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