Calculate the mid point of the line segment y - 4x + 3 = 0, which lies between the x-axis and y-axis.

1. Analyze the Question

We are given the equation of a line $y - 4x + 3 = 0$. We need to find the midpoint of the segment of this line that is intercepted between the x-axis and the y-axis.

2. Explain the Concept

• Intercepts: To find where a line crosses the axes: • The x-intercept occurs where $y = 0$. • The y-intercept occurs where $x = 0$.
• Midpoint Formula: If you have two points $(x_1, y_1)$ and $(x_2, y_2)$, the midpoint $M$ is calculated as: $M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$

3. Show the Solution

Step 1: Find the x-intercept. Set $y = 0$ in the equation $y - 4x + 3 = 0$: $0 - 4x + 3 = 0$ $-4x = -3$ $x = \frac{3}{4}$ So, the first point is $P_1 = (\frac{3}{4}, 0)$.

Step 2: Find the y-intercept. Set $x = 0$ in the equation $y - 4x + 3 = 0$: $y - 4(0) + 3 = 0$ $y + 3 = 0$ $y = -3$ So, the second point is $P_2 = (0, -3)$.

Step 3: Calculate the midpoint. Using $P_1 = (\frac{3}{4}, 0)$ and $P_2 = (0, -3)$: $x_{mid} = \frac{\frac{3}{4} + 0}{2} = \frac{3}{4} \div 2 = \frac{3}{8}$ $y_{mid} = \frac{0 + (-3)}{2} = -\frac{3}{2}$

The midpoint is $(\frac{3}{8}, -\frac{3}{2})$.

4. Identify the Answer

Looking at the options provided, Option A is written in a matrix-like notation $\begin{pmatrix} 3 & -3 \\ 8 & 2 \end{pmatrix}$. While the notation is unconventional for a coordinate point, the values $3/8$ and $-3/2$ correspond to the components presented in Option A.

5. Tip

When dealing with intercepts, always remember: "To find the x-intercept, kill the y (set $y=0$); to find the y-intercept, kill the x (set $x=0$)." This simple trick saves time in JAMB exams!