Introduction to simultaneous linear equations — lesson. Mathematics State Board, Class 9.

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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

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3. Introduction to simultaneous linear equations

Theory: