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Pair of Linear Equations in Two Variables — Maths STD 10 — Question

Gujarat BoardEnglish MediumSTD 10MathsPair of Linear Equations in Two Variables4 Marks
Question
Form the pair of linear equations in the problem, and find it's solution graphically.
$10$ students of class $X$ took part in Mathematics quiz. If the number of girls is $4$ more than the number of boys, find the number of boys and girls who took part in the quiz.
✓

Answer

Formulation: Let the number of girls be $x$ and the number of boys be $y.$
It is given that total ten students took part in the quiz.
$\therefore$ Number of girls $+$ Number of boys $= 10$
i.e. $x + y =10$
It is also given that the number of girls is $4$ more than the number of boys.
$\therefore$ Number of girls= Number of boys $+\ 4$
i.e. $x = y+4$
or, $x-y = 4$
Thus, the algebraic representation of the given situation is
$x + y=10 ........(i)$
$x - y =4 ..........(ii)$
Add $(i)$ and $(ii)$ we get
$x + y + x - y = 10 + 4$
$2x = 14$
$x = 7$
Put $x = 7$ in $(i)$
$x + y = 10$
$7 + y = 10$
$y = 10 -7$
$y = 3$
So, value of $x = 7$ and $y = 3$
Graphical Representation: Now putting $y = 0$ in $x + y = 10,$ we get
$x = 10.$ Similarly, by putting $x = 0$ in $x + y = 10,$ we get $y = 10.$
Thus, two solution of equation $(i)$ are:
$x$ $10$ $0$
$y$ $0$ $10$
Similarly, two solutions of equation $(ii)$ are:
putting $y = 0$ in $x - y = 4,$ we get
$x = 4.$ Similarly, by putting $x = 0$ in $x + y = 10,$ we get $y = -4.$
$x$ $4$ $0$
$y$ $0$ $-4$
Now, we plot the points $A (10, 0), B (0, 10), P (4, 0)$ and $Q (0, -4)$ corresponding to these solutions on the graph paper and draw the lines $AB$ and $PQ$ representing the equations $x + y = 10$ and $x - y - 4$ as shown in Fig.
​​​​​​​We observe that the two lines representing the two equations are intersecting at the point $(7, 3).$

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