Find the equation of a line perpendicular to th... - JAMB Mathematics 2024 Question

We are tasked with finding the equation of a line that is perpendicular to the line given by 4y = 7x + 3 and passes through the point (-3, 1).

Step 1: Rewrite the given line in slope-intercept form (y = mx + c)

Start by dividing through by 4 to isolate y:

\( 4y = 7x + 3 \)

\( y = \frac{7}{4}x + \frac{3}{4} \)

From this, the slope \( m_1 \) of the given line is \( \frac{7}{4} \).

Step 2: Find the slope of the perpendicular line

The slope of a line perpendicular to another is the negative reciprocal of the original slope. Therefore, the slope \( m_2 \) of the perpendicular line is:

\( m_2 = -\frac{1}{m_1} = -\frac{1}{\frac{7}{4}} = -\frac{4}{7} \).

Step 3: Use the point-slope form of a line equation

The equation of a line with slope \( m \) passing through a point \((x_1, y_1)\) is:

\( y - y_1 = m(x - x_1) \)

Substitute \( m_2 = -\frac{4}{7} \) and the point (-3, 1):

\( y - 1 = -\frac{4}{7}(x + 3) \)

Step 4: Simplify the equation

Expand the right-hand side:

\( y - 1 = -\frac{4}{7}x - \frac{12}{7} \)

Combine like terms:

\( y = -\frac{4}{7}x - \frac{12}{7} + 1 \)

\( y = -\frac{4}{7}x - \frac{12}{7} + \frac{7}{7} \)

\( y = -\frac{4}{7}x - \frac{5}{7} \)

Step 5: Eliminate fractions to write in general form

Multiply through by 7 to eliminate fractions:

\( 7y = -4x - 5 \)

Rearrange to general form:

\( 4x + 7y + 5 = 0 \)

Final Answer: 7y + 4x + 5 = 0