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

Step 1: Write both lines in slope-intercept form.
For two lines to be parallel, their slopes must be equal.
The first line is \(4x + 7y = 6\), so
\[ y = -\frac{4}{7}x + \frac{6}{7} \]
This gives slope \(m_1 = -\dfrac{4}{7}\).

Step 2: Find the slope of the second line.
The second line is \(3ax + 42y = 24\), so
\[ y = -\frac{3a}{42}x + \frac{24}{42} = -\frac{a}{14}x + \frac{4}{7} \]
This gives slope \(m_2 = -\dfrac{a}{14}\).

Step 3: Set the two slopes equal.
Since the lines are parallel, \(m_1 = m_2\):
\[ -\frac{4}{7} = -\frac{a}{14} \]
\[ \frac{a}{14} = \frac{4}{7} \]

Step 4: Solve for \(a\).
\[ a = 14 \times \frac{4}{7} = \frac{56}{7} = 8 \]

Final Answer:
The value of \(a\) that makes the two lines parallel is
\[ \boxed{a = 8} \]