Question

Solve the following system of equations graphically: x - 2y = 3, 3x - 8y = 7.

Hint:

To verify your graph under time constraints, solve the system algebraically using substitution or elimination first.
This takes only a few seconds and gives you the exact intersection point, ensuring you plot and label the graph perfectly.

Answer 1

Step 1: Understanding the Question:
The topic of this question is Pair of Linear Equations in Two Variables (Graphical Solution).
We are given two linear equations: \( x - 2y = 3 \) and \( 3x - 8y = 7 \).
We need to plot both equations on a coordinate grid and locate their intersection point, which gives the unique solution to the system.

Step 2: Key Formula or Approach:
- Convert each equation to express \( x \) in terms of \( y \) (or vice versa) to find coordinate pairs.
- Plot at least two points for each line and draw the straight lines.
- The point of intersection \( (x, y) \) is the solution to the system of equations.

Step 3: Detailed Explanation:
1. Analyze Equation 1:
\[ x - 2y = 3 \implies x = 2y + 3 \]
Find three coordinate points by substituting values for \( y \):
- If \( y = 0 \): \( x = 2(0) + 3 = 3 \). Point \( P_1 = (3, 0) \).
- If \( y = -1 \): \( x = 2(-1) + 3 = 1 \). Point \( P_2 = (1, -1) \).
- If \( y = 1 \): \( x = 2(1) + 3 = 5 \). Point \( P_3 = (5, 1) \).
2. Analyze Equation 2:
\[ 3x - 8y = 7 \implies 3x = 8y + 7 \implies x = \frac{8y + 7}{3} \]
Find three coordinate points by substituting integer-yielding values for \( y \):
- If \( y = 1 \): \( x = \frac{8(1) + 7}{3} = \frac{15}{3} = 5 \). Point \( Q_1 = (5, 1) \).
- If \( y = -2 \): \( x = \frac{8(-2) + 7}{3} = \frac{-9}{3} = -3 \). Point \( Q_2 = (-3, -2) \).
- If \( y = 4 \): \( x = \frac{8(4) + 7}{3} = \frac{39}{3} = 13 \). Point \( Q_3 = (13, 4) \).
3. Plot the graph:
- Draw the Cartesian coordinate axes \( XOX' \) and \( YOY' \).
- Plot points \( (3, 0) \), \( (1, -1) \), and \( (5, 1) \) on the grid and join them with a straight line. This represents the equation \( x - 2y = 3 \).
- Plot points \( (5, 1) \) and \( (-3, -2) \) on the same grid and join them with another straight line. This represents the equation \( 3x - 8y = 7 \).
4. Identify the intersection point:
Observe the graph to find where the two lines cross.
The two lines intersect exactly at point \( (5, 1) \).
Verify the point:
- For line 1: \( 5 - 2(1) = 3 \) (True).
- For line 2: \( 3(5) - 8(1) = 15 - 8 = 7 \) (True).

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