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: Build the table of points for the first line using a different set of y-values.
First equation: $x - 2y = 3 \implies x = 2y + 3$.
- If $y = -3 \implies x = 2(-3)+3 = -3$. Point: $(-3,-3)$
- If $y = 2 \implies x = 2(2)+3 = 7$. Point: $(7,2)$
- If $y = 6 \implies x = 2(6)+3 = 15$. Point: $(15,6)$

Step 2: Build the table of points for the second line using a different set of y-values.
Second equation: $3x - 8y = 7 \implies x = \frac{8y+7}{3}$.
- If $y = -5 \implies x = \frac{8(-5)+7}{3} = \frac{-33}{3} = -11$. Point: $(-11,-5)$
- If $y = 7 \implies x = \frac{8(7)+7}{3} = \frac{63}{3} = 21$. Point: $(21,7)$
- If $y = 10 \implies x = \frac{8(10)+7}{3} = \frac{87}{3} = 29$. Point: $(29,10)$

Step 3: Plot both lines using these points and read off the intersection.
Plotting $(-3,-3), (7,2), (15,6)$ gives the straight line for the first equation. Plotting $(-11,-5), (21,7), (29,10)$ gives the straight line for the second equation. The two lines cross at the point $(5, 1)$.

Step 4: Confirm the graphical answer using a quick algebraic elimination check.
Multiply the first equation by $4$:
\[ 4x - 8y = 12 \]
Subtract the second equation $3x - 8y = 7$ from this:
\[ (4x - 8y) - (3x - 8y) = 12 - 7 \]
\[ x = 5 \]
Substitute $x = 5$ into $x - 2y = 3$:
\[ 5 - 2y = 3 \implies 2y = 2 \implies y = 1 \]
This matches the graphical intersection point exactly.

Final Answer:
The solution of the system is $(x, y) = (5, 1)$. \[ \boxed{(x,y) = (5,1)} \]