Question

Use graphical method to solve the system of linear equations : $y = -3$ and $x + 2y = 4$.

Hint:

To find the intersection coordinate quickly without guessing, substitute $y = -3$ directly into the second equation:
\[ x + 2(-3) = 4 \implies x - 6 = 4 \implies x = 10 \]
Plot this specific point $(10, -3)$ to guarantee your graph is perfectly accurate!

Answer 1

Step 1: Understanding the Question:
We are given a system of two linear equations in two variables:
1. $y = -3$
2. $x + 2y = 4$
We need to solve this system graphically by plotting both lines on a Cartesian coordinate plane and identifying their unique point of intersection.

Step 2: Key Formula or Approach:
- The equation $y = c$ represents a horizontal line parallel to the $x$-axis, passing through the point $(0, c)$.
- For the second equation, $x + 2y = 4$, we can find distinct points by substituting arbitrary values of $y$ and calculating corresponding $x$ values, then plotting them to draw a straight line.
- The point of intersection of both lines represents the unique solution $(x, y)$.

Step 3: Detailed Explanation:

• Plot the first line, $y = -3$:
This is a horizontal line parallel to the $x$-axis, passing through $(0, -3)$.
Some coordinate points on this line are:
\[ P_1(0, -3), \quad P_2(4, -3), \quad P_3(10, -3) \]

• Find coordinate points for the second line, $x + 2y = 4$:
Express $x$ in terms of $y$:
\[ x = 4 - 2y \]
Let us calculate some coordinates:
- When $y = 0$:
\[ x = 4 - 2(0) = 4 \implies (4, 0) \]
- When $y = 2$:
\[ x = 4 - 2(2) = 0 \implies (0, 2) \]
- When $y = -3$ (the value of our first line):
\[ x = 4 - 2(-3) = 4 + 6 = 10 \implies (10, -3) \]

• Plot the graph:
- Draw the Cartesian $x$ and $y$ axes.
- Draw the horizontal line $y = -3$.
- Plot the points $(4, 0)$, $(0, 2)$, and $(10, -3)$, and join them with a straight line representing $x + 2y = 4$.

• Identify the intersection point:
- Observe where both lines intersect.
- The lines cross exactly at the point $(10, -3)$.


Step 4: Final Answer:
The graphical solution of the system of equations is $x = 10$ and $y = -3$.