Form the pair of linear equations in the following problem and find their solution graphically:
$10$ students of Class X took part in a 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. Also find the coordinates of the points where these lines intersect the $y$-axis.
Step-by-Step Solution
Key Concept: Let number of girls be $x$ and boys be $y$. Equations: $x + y = 10$ and $x - y = 4$. Plot both lines and find intersection point.
Stepwise Solution:
Let number of girls $= x$ and boys $= y$. System: $x + y = 10$ -- (Line 1) and $x - y = 4$ -- (Line 2). [1.0 Mark]
Table of values:
Line 1 ($x + y = 10$): Points $(0, 10), (7, 3), (10, 0)$.
Line 2 ($x - y = 4$): Points $(4, 0), (7, 3), (0, -4)$. [1.5 Marks]
Graph plotting: Both lines intersect at point $(7, 3)$.
Hence, number of girls $x = 7$ and number of boys $y = 3$. [1.5 Marks]
$y$-intercepts: Line 1 intersects $y$-axis at $(0, 10)$; Line 2 intersects $y$-axis at $(0, -4)$. [1.0 Mark]
Marking Scheme:
• Forming pair of linear equations: 1.0 Mark
• Preparing accurate tables of values for both lines: 1.5 Marks
• Plotting lines and identifying intersection point (7, 3) -> 7 girls, 3 boys: 1.5 Marks
• Stating $y$-axis intersection points $(0, 10)$ and $(0, -4)$: 1.0 Mark
Correct Answer: