Question

Represent the following pair of linear equations graphically and hence comment on the condition of consistency of this pair : \(x - 5y = 6; 2x - 10y = 12\)

Hint:

Checking the coefficient ratio condition first only takes a few seconds and provides a blueprint of what your graph should look like.
If the ratios are all equal, expect your lines to lie directly on top of each other!

Answer 1

Step 1: Understanding the Question:
The problem requires us to graph a given pair of linear equations on a Cartesian plane and analyze their geometric relationship to determine the consistency of the system.

Step 2: Key Formula or Approach:
1. Find coordinating points \((x, y)\) for each equation to draw the lines.
2. Check the ratio of coefficients to analytically confirm the graph's outcome:
\[ \frac{a_1}{a_2} = \frac{1}{2}, \quad \frac{b_1}{b_2} = \frac{-5}{-10} = \frac{1}{2}, \quad \frac{c_1}{c_2} = \frac{6}{12} = \frac{1}{2} \]
3. Since \(\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}\), the equations represent coincident lines.

Step 3: Detailed Explanation:
1. Let the first equation be:
\[ x - 5y = 6 \implies x = 5y + 6 \]
- If \(y = 0 \implies x = 6\). Point \(A(6, 0)\).
- If \(y = -1 \implies x = 1\). Point \(B(1, -1)\).
- If \(y = -2 \implies x = -4\). Point \(C(-4, -2)\).
2. Let the second equation be:
\[ 2x - 10y = 12 \implies x - 5y = 6 \]
- Dividing by 2 shows this equation is mathematically identical to the first.
- Plotting the coordinates of both equations will result in the exact same line on the graph paper.
3. Geometrical interpretation:
- When plotted, the two lines overlap completely (coincident lines).
- This means there are infinitely many points of intersection.
- A system of linear equations is consistent if it has at least one solution. Since there are infinite solutions, the system is consistent and dependent.

Step 4: Final Answer:
The lines are coincident. The pair of linear equations is consistent and dependent.