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Pair of Linear Functions in Two Variables

    Graphical Method of Solution of a Pair of Linear Equations

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10th class > Pair of Linear Functions in Two Variables > Graphical Method of Solution of a Pair of Linear Equations

Graphical Method of Solution of a Pair of Linear Equations

A pair of linear equations which has no solution is called an pair of linear equations.

A pair of linear equations in two variables which has a solution, is called a pair of linear equations.

A pair of linear equations which are has infinitely many distinct common solutions. Such a pair is called a pair of linear equations in two variables.

Note: A dependent pair of linear equations is always consistent.

We can now summarise the behaviour of lines representing a pair of linear equations in two variables and the existence of solutions as follows:

(i) If the lines may intersect in a single point. In this case, the pair of equations has a solution(s). They are also a consistent pair of equations.

(ii) If the lines may be parallel. In this case, the equations have solution(s). They are also an inconsistent pair of equations.

(iii) If the lines may be coincident, the equations have solutions. They are also and pair of equations.

Consider the following three pairs of equations.

(i) x – 2y = 0 and 3x + 4y – 20 = 0 (The lines intersect)

(ii) 2x + 3y – 9 = 0 and 4x + 6y – 18 = 0 (The lines coincide)

(iii) x + 2y – 4 = 0 and 2x + 4y – 12 = 0 (The lines are parallel)

Let us now write down, and compare, the values of a1a2 = b1b2 and c1c2 in all of the three above equation.

Here, a1,a2, b1,b2, and c1 c2 all denote the of equations given in the general form.
Pair of Linesa1a2b1b2c1c2Ratio ComaprisonGraphical RepresentationInterpretation
x+2y=0, 3x+4y–20=013240−20a1a2 b1b2Intersecting linesExactly one(unique) solution
2x+3y–9=0, 4x+6y–18=02436−9−18a1a2 b1b2 c1c2Coincident linesInfinitely many solutions
x+2y–4=0, 2x+4y–12=01224−4−12a1a2 b1b2 c1c2Parallel linesNo solutions

From the table above, you can observe that if the lines represented by the equation:

a1x + b1y + c1 = 0

a2x + b2y + c2 = 0

are (i) intersecting, then a1a2 ≠ b1b2.

(ii) coincident, then a1a2 = b1b2 = c1c2.

(iii) parallel, then a1a2 = b1b2 ≠ c1c2

Note: In fact, the converse is also true for pair of lines.

Example 1:Check graphically whether the pair of equations

x + 3y = 6 (1) and 2x – 3y = 12 (2)

is consistent. If so, solve them graphically.

For: x + 3y = 6

xy
0
-3
-6
6

For: 2x – 3y = 12

xy
0
3
-3
-6

Thus, the solution is

In the above graph we can see this solution.

Graphically, find whether the following pair of equations has no solution, unique solution or infinitely many solutions:

5x – 8y + 1 = 0 (1)

3x – 245 y + 35 = 0 (2)

Solution: Multiplying Equation(2) by 53 we get: 5x – 8y + 1 = 0

This is the same as Equation (1). Hence the lines represented by Equations (1) and (2) are coincident and have solutions.

Plotting on the graph, we get:

In the above graph we can see that both the equation lines coincident.

👋 Well done...

Champa went to a ‘Sale’ to purchase some pants and skirts. When her friends asked her how many of each she had bought, she answered, “The number of skirts is two less than twice the number of pants purchased. Also, the number of skirts is four less than four times the number of pants purchased”. Help her friends to find how many pants and skirts Champa bought.

y = 2x – 2xy
2
0
y = 4x – 4xy
0
1

Thus, the solution is

In the above graph we can see this solution.

👍 Great...