Question

If the straight lines given by the following two equations are parallel to each other, the value of \(a\) is _____. (answer in integer) \[ 4x + 7y = 6; \quad 3ax + 42y = 24 \]

Hint:

Two lines \(a_1x+b_1y=c_1\) and \(a_2x+b_2y=c_2\) are parallel when \(a_1/a_2 = b_1/b_2\); apply this to find \(a\).

Answer 1

Correct Answer: 8

Instead of converting each line to $y = mx + c$ form, this method uses the direct coefficient test for two lines $a_1x + b_1y = c_1$ and $a_2x + b_2y = c_2$ to be parallel: they are parallel exactly when $\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2}$ (and this common ratio differs from $c_1/c_2$, otherwise the lines would coincide).

Here the two lines are $4x + 7y = 6$ and $3ax + 42y = 24$. Matching coefficients:

• $a_1 = 4$, $b_1 = 7$, $c_1 = 6$
• $a_2 = 3a$, $b_2 = 42$, $c_2 = 24$

Apply the parallel condition $\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2}$:

$$ \frac{4}{3a} = \frac{7}{42} $$

Simplify the right side first: $7/42 = 1/6$. So the equation becomes:

$$ \frac{4}{3a} = \frac{1}{6} $$

Cross multiply:

$$ 4 \times 6 = 3a \times 1 $$$$ 24 = 3a $$$$ a = 8 $$

As a quick check, substitute $a = 8$ back: the second line becomes $24x + 42y = 24$, which simplifies (dividing by 6) to $4x + 7y = 4$. This has the same $x$ and $y$ coefficients as the first line $4x + 7y = 6$, only the constant term differs, confirming the two lines are parallel and distinct (not the same line).

Let's summarize:

• Two lines $a_1x+b_1y=c_1$ and $a_2x+b_2y=c_2$ are parallel when $a_1/a_2 = b_1/b_2$.
• Applying this ratio test directly gives $a = 8$ without needing to rearrange either equation into slope form.

So the value of $a$ is $8$.