Question

Solve any one of the following internal choices (a) or (b):
34(a) Solve the following system of equations graphically : $x - 2y = 3, 3x - 8y = 7$

Hint:

When choosing points for fractional equations like $x = \frac{8y + 7}{3}$, pick values of $y$ that make the numerator a multiple of 3 so that you obtain integer values for $x$, making them easy to plot on standard graph paper!

Answer 1

Step 1: Understanding the Question:
This question is from "Pair of Linear Equations in Two Variables".
We are given two linear equations and need to solve them graphically.
By finding several coordinates for each line, plotting them on a graph, and drawing the straight lines, we can identify their point of intersection, which represents the common solution to both equations.

Step 2: Key Formula or Approach:
1. Express $x$ in terms of $y$ (or vice versa) for each equation.
2. Generate tables of coordinates by substituting convenient values of $y$ to find corresponding values of $x$.
3. Plot the points on graph paper, draw the straight lines, and find the intersection point $(x, y)$.

Step 3: Detailed Explanation:

• First Equation: $x - 2y = 3 \implies x = 2y + 3$
- If $y = 0 \implies x = 2(0) + 3 = 3$. Point: $(3, 0)$
- If $y = -1 \implies x = 2(-1) + 3 = 1$. Point: $(1, -1)$
- If $y = 1 \implies x = 2(1) + 3 = 5$. Point: $(5, 1)$

• Second Equation: $3x - 8y = 7 \implies 3x = 8y + 7 \implies x = \frac{8y + 7}{3}$
- If $y = -2 \implies x = \frac{8(-2) + 7}{3} = \frac{-9}{3} = -3$. Point: $(-3, -2)$
- If $y = 1 \implies x = \frac{8(1) + 7}{3} = \frac{15}{3} = 5$. Point: $(5, 1)$
- If $y = 4 \implies x = \frac{8(4) + 7}{3} = \frac{39}{3} = 13$. Point: $(13, 4)$

• Plotting and Intersection:
- Draw horizontal (x-axis) and vertical (y-axis) coordinate lines on graph paper.
- Plot the points $(3,0), (1,-1), (5,1)$ and draw a straight line through them representing $x - 2y = 3$.
- Plot the points $(-3,-2), (5,1), (13,4)$ and draw a straight line through them representing $3x - 8y = 7$.
- Observe that both straight lines intersect at the point $(5, 1)$.

• Verification:
- Substituting $x=5, y=1$ in first equation: $5 - 2(1) = 3$ (True)
- Substituting in second equation: $3(5) - 8(1) = 15 - 8 = 7$ (True)

Step 4: Final Answer:
The graphical solution of the system of linear equations is $(x, y) = (5, 1)$.