The gradient of a line joining (x,4) and (1,2) is $\frac{1}{2}$. Find the value of x

Analysis

To find the value of $x$, we need to use the formula for the gradient (slope) of a straight line passing through two points $(x_1, y_1)$ and $(x_2, y_2)$.

Concept

The gradient ($m$) of a line is defined as the change in $y$ divided by the change in $x$:

In this problem, we are given:

• Point 1: $(x_1, y_1) = (x, 4)$
• Point 2: $(x_2, y_2) = (1, 2)$
• Gradient ($m$) = $\frac{1}{2}$

Solution

1. Substitute the given values into the gradient formula: $$\frac{1}{2} = \frac{2 - 4}{1 - x}$$

2. Simplify the numerator: $$\frac{1}{2} = \frac{-2}{1 - x}$$

3. Cross-multiply to solve for $x$: $$1(1 - x) = 2(-2)$$ $$1 - x = -4$$

4. Isolate $x$: $$-x = -4 - 1$$ $$-x = -5$$ $$x = 5$$

Conclusion

By substituting the coordinates into the gradient formula and solving the resulting linear equation, we find that $x = 5$.

Exam Tip: When dealing with gradient problems, always be careful with the signs of the coordinates. A common mistake is to subtract the coordinates in the wrong order (e.g., $y_1 - y_2$ on top but $x_2 - x_1$ on the bottom). Always maintain the same order for both: $\frac{y_2 - y_1}{x_2 - x_1}$ or $\frac{y_1 - y_2}{x_1 - x_2}$.