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: Simplify the second equation directly, instead of computing three separate coefficient ratios.
The equations are:
\[ x-5y=6 \qquad \text{...(i)} \]
\[ 2x-10y=12 \qquad \text{...(ii)} \]
Every term in (ii) is exactly double the corresponding term in (i). Dividing (ii) by 2:
\[ \frac{2x}{2}-\frac{10y}{2} = \frac{12}{2} \implies x-5y=6 \]
This is identical to (i), so without computing any coefficient ratios we can see both equations describe the same line.

Step 2: Find points to plot this one line.
Using $x-5y=6$, that is $x=5y+6$: if $y=0$, $x=6$, point $(6,0)$; if $y=-1$, $x=1$, point $(1,-1)$; if $y=-2$, $x=-4$, point $(-4,-2)$.

Step 3: Describe the graph.
Plotting $(6,0)$, $(1,-1)$, $(-4,-2)$ and joining them gives one straight line. Since equation (ii) reduces to the same equation as (i), plotting it would trace out exactly the same line rather than a second, separate one.

Step 4: Comment on consistency.
Because the two lines coincide completely, every point on the line is a common solution to both equations, giving infinitely many solutions. A pair of equations with at least one solution is consistent, and when it has infinitely many solutions because the lines coincide, it is further called dependent.

Final Answer:
The two lines coincide, so the pair of equations is consistent and dependent, with infinitely many common solutions.