- Identify the slope formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\) for points \(P(0, 1)\) and \(Q(4, y_2)\) with slope \(m = \frac{3}{4}\).
- Substitute the given values into the slope equation: \(\frac{3}{4} = \frac{y_2 - 1}{4 - 0}\).
- Solve for \(y_2\) by simplifying and isolating the variable: \(y_2 = 4\).
- The missing coordinate for point \(Q\) is \(\boxed{4}\).
Explanation
- Setting Up the Slope Equation
The slope of a line passing through two points \(P(x_1, y_1)\) and \(Q(x_2, y_2)\) is given by the formula:
\[m = \frac{y_2 - y_1}{x_2 - x_1}\]
We are given that point \(P\) is \((0, 1)\), which means \(x_1 = 0\) and \(y_1 = 1\). Point \(Q\) is given as \((4, y_2)\), which means \(x_2 = 4\) and \(y_2\) is the value we need to find. We are also given that the line \(PQ\) is parallel to a line with a slope of \(\frac{3}{4}\), meaning the slope \(m\) for line \(PQ\) is also \(\frac{3}{4}\).
- Solving for the Unknown Coordinate
Substitute the known values \(m = \frac{3}{4}\), \(x_1 = 0\), \(y_1 = 1\), and \(x_2 = 4\) into the slope formula:
\[\frac{3}{4} = \frac{y_2 - 1}{4 - 0}\]
This simplifies to:
\[\frac{3}{4} = \frac{y_2 - 1}{4}\]
To solve for \(y_2\), multiply both sides of the equation by 4:
\[4 \times \left(\frac{3}{4}\right) = 4 \times \left(\frac{y_2 - 1}{4}\right)\]
\[3 = y_2 - 1\]
Adding 1 to both sides, we get:
\[y_2 = 3 + 1 = 4\]
Thus, the coordinate for point \(Q\) is \((4, 4)\).
Examples
In urban planning, architects often use slopes to design accessible ramps. If a city regulation requires a wheelchair ramp to have a slope of \(\frac{3}{4}\) (or 75% grade), and the ramp starts at the origin \((0, 0)\) of a coordinate system, the vertical height at a horizontal distance of 4 meters can be found using the same slope formula. Understanding how slope relates to the change in coordinates helps in construction, road design, and various engineering projects to ensure stability and accessibility.