Question

Use graphical method to solve the system of linear equations : $x = -3$ and $5x - 2y = -5$.

Hint:

To find the intersection coordinate quickly without guessing, substitute $x = -3$ directly into the second equation:
\[ 5(-3) - 2y = -5 \implies -15 - 2y = -5 \implies 2y = -10 \implies y = -5 \]
Plot this specific point $(-3, -5)$ 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. $x = -3$
2. $5x - 2y = -5$
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 $x = c$ represents a vertical line parallel to the $y$-axis, passing through the point $(c, 0)$.
- For the second equation, $5x - 2y = -5$, we can find distinct points by substituting arbitrary values of $x$ and calculating corresponding $y$ 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, $x = -3$:
This is a vertical line parallel to the $y$-axis, passing through $(-3, 0)$.
Some coordinate points on this line are:
\[ P_1(-3, 0), \quad P_2(-3, -5), \quad P_3(-3, 5) \]

• Find coordinate points for the second line, $5x - 2y = -5$:
Express $2y$ in terms of $x$:
\[ 2y = 5x + 5 \implies y = \frac{5x + 5}{2} \]
Let us calculate some coordinates:
- When $x = -1$:
\[ y = \frac{5(-1) + 5}{2} = 0 \implies (-1, 0) \]
- When $x = 1$:
\[ y = \frac{5(1) + 5}{2} = 5 \implies (1, 5) \]
- When $x = -3$ (the value of our first line):
\[ y = \frac{5(-3) + 5}{2} = \frac{-10}{2} = -5 \implies (-3, -5) \]

• Plot the graph:
- Draw the Cartesian $x$ and $y$ axes.
- Draw the vertical line $x = -3$.
- Plot the points $(-1, 0)$, $(1, 5)$, and $(-3, -5)$, and join them with a straight line representing $5x - 2y = -5$.

• Identify the intersection point:
- Observe where both lines intersect.
- The lines cross exactly at the point $(-3, -5)$.


Step 4: Final Answer:
The graphical solution of the system of equations is $x = -3$ and $y = -5$.