Graphical Method of Solution of a Pair of Linear Equations
Let us look at the earlier examples one by one.
♦ In the situation of Example 1, find out how many rides on the Giant Wheel Akhila had, and how many times she played Hoopla.
In Fig. 3.2, you noted that the equations representing the situation are geometrically shown by two lines intersecting at the point `(4, 2).` Therefore, the point `(4, 2)` lies on the lines represented by both the equations `x – 2y = 0` and `3x + 4y = 20.` And this is the only common point.
Let us verify algebraically that x = 4, y = 2 is a solution of the given pair of equations. Substituting the values of x and y in each equation, we get `4 – 2 × 2 = 0` and `3(4) + 4(2) = 20.` So, we have verified that `x = 4, y = 2` is a solution of both the equations.
`=> "Since (4, 2) is the only common point on both the lines, "`
`"there is one and only one solution for this pair of linear equations in two variables."`
Thus, the number of rides Akhila had on Giant Wheel is `4` and the number of times she played Hoopla is `2.`
♦ In the situation of Example 2, can you find the cost of each pencil and each eraser?
`=>` In Fig. 3.3, the situation is geometrically shown by a pair of coincident lines. The solutions of the equations are given by the common points.
`=>` From the graph, we observe that every point on the line is a common solution to both the equations. So, the equations `2x + 3y = 9` and `4x + 6y = 18` have infinitely many solutions. This should not surprise us, because if we divide the equation 4x + 6y = 18 by 2 , we get `2x + 3y = 9,` which is the same as Equation (1). That is, both the equations are equivalent. From the graph, we see that any point on the line gives us a possible cost of each pencil and eraser. For instance, each pencil and eraser can cost Rs. 3 and Rs. 1 respectively. Or, each pencil can cost Rs. 3.75 and eraser can cost Rs. 0.50, and so on.
♦ In the situation of Example 3, can the two rails cross each other?
`=>` In Fig. 3.4, the situation is represented geometrically by two parallel lines. Since the lines do not intersect at all, the rails do not cross. This also means that the equations have no common solution.
`=>` A pair of linear equations which has no solution, is called an inconsistent pair of linear equations. A pair of linear equations in two variables, which has a solution, is called a consistent pair of linear equations. A pair of linear equations which are equivalent has infinitely many distinct common solutions. Such a pair is called a dependent pair of linear equations in two variables. Note that a dependent pair of linear equations is always consistent.
`=>` We can now summaries the behavior of lines representing a pair of linear equations in two variables and the existence of solutions as follows :
(i) the lines may intersect in a single point. In this case, the pair of equations has a unique solution (consistent pair of equations).
(ii) the lines may be parallel. In this case, the equations have no solution (inconsistent pair of equations).
(iii) the lines may be coincident. In this case, the equations have infinitely many
solutions [dependent (consistent) pair of equations].
`=>` Let us now go back to the pairs of linear equations formed in Examples 1, 2, and 3, and note down what kind of pair they are geometrically.
(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 ` (a_1)/(a_2) , (b_1)/(b_2) ` and ` (c_1)/(C_2)` in all the three examples. Here, `a_1, b_1, c_1` and `a_2, b_2, c_2` denote the coefficents of equations given in the general form in Section 3.2.
`=>` From the table above, you can observe that if the lines represented by the equation
`a_1 x + b_1 y + c_1 = 0`
and `a_2 x + b_2 y + c_2 = 0`
are (i) intersecting, then `color{orange}{a_1/a_2 ≠ b_1/b_2}` .
(ii) coincident, then ` color{orange}{a_1/a_2 = b_1/b_2 =c_1/c_2}` .
(iii) parallel, then ` color{orange}{a_1/a_2 = b_1/b_2 ≠ c_1/c_2}`.
In fact, the converse is also true for any pair of lines. You can verify them by considering some more examples by yourself.
♦ In the situation of Example 1, find out how many rides on the Giant Wheel Akhila had, and how many times she played Hoopla.
In Fig. 3.2, you noted that the equations representing the situation are geometrically shown by two lines intersecting at the point `(4, 2).` Therefore, the point `(4, 2)` lies on the lines represented by both the equations `x – 2y = 0` and `3x + 4y = 20.` And this is the only common point.
Let us verify algebraically that x = 4, y = 2 is a solution of the given pair of equations. Substituting the values of x and y in each equation, we get `4 – 2 × 2 = 0` and `3(4) + 4(2) = 20.` So, we have verified that `x = 4, y = 2` is a solution of both the equations.
`=> "Since (4, 2) is the only common point on both the lines, "`
`"there is one and only one solution for this pair of linear equations in two variables."`
Thus, the number of rides Akhila had on Giant Wheel is `4` and the number of times she played Hoopla is `2.`
♦ In the situation of Example 2, can you find the cost of each pencil and each eraser?
`=>` In Fig. 3.3, the situation is geometrically shown by a pair of coincident lines. The solutions of the equations are given by the common points.
`=>` From the graph, we observe that every point on the line is a common solution to both the equations. So, the equations `2x + 3y = 9` and `4x + 6y = 18` have infinitely many solutions. This should not surprise us, because if we divide the equation 4x + 6y = 18 by 2 , we get `2x + 3y = 9,` which is the same as Equation (1). That is, both the equations are equivalent. From the graph, we see that any point on the line gives us a possible cost of each pencil and eraser. For instance, each pencil and eraser can cost Rs. 3 and Rs. 1 respectively. Or, each pencil can cost Rs. 3.75 and eraser can cost Rs. 0.50, and so on.
♦ In the situation of Example 3, can the two rails cross each other?
`=>` In Fig. 3.4, the situation is represented geometrically by two parallel lines. Since the lines do not intersect at all, the rails do not cross. This also means that the equations have no common solution.
`=>` A pair of linear equations which has no solution, is called an inconsistent pair of linear equations. A pair of linear equations in two variables, which has a solution, is called a consistent pair of linear equations. A pair of linear equations which are equivalent has infinitely many distinct common solutions. Such a pair is called a dependent pair of linear equations in two variables. Note that a dependent pair of linear equations is always consistent.
`=>` We can now summaries the behavior of lines representing a pair of linear equations in two variables and the existence of solutions as follows :
(i) the lines may intersect in a single point. In this case, the pair of equations has a unique solution (consistent pair of equations).
(ii) the lines may be parallel. In this case, the equations have no solution (inconsistent pair of equations).
(iii) the lines may be coincident. In this case, the equations have infinitely many
solutions [dependent (consistent) pair of equations].
`=>` Let us now go back to the pairs of linear equations formed in Examples 1, 2, and 3, and note down what kind of pair they are geometrically.
(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 ` (a_1)/(a_2) , (b_1)/(b_2) ` and ` (c_1)/(C_2)` in all the three examples. Here, `a_1, b_1, c_1` and `a_2, b_2, c_2` denote the coefficents of equations given in the general form in Section 3.2.
`=>` From the table above, you can observe that if the lines represented by the equation
`a_1 x + b_1 y + c_1 = 0`
and `a_2 x + b_2 y + c_2 = 0`
are (i) intersecting, then `color{orange}{a_1/a_2 ≠ b_1/b_2}` .
(ii) coincident, then ` color{orange}{a_1/a_2 = b_1/b_2 =c_1/c_2}` .
(iii) parallel, then ` color{orange}{a_1/a_2 = b_1/b_2 ≠ c_1/c_2}`.
In fact, the converse is also true for any pair of lines. You can verify them by considering some more examples by yourself.
