A line that includes the points $(-4, j)$ and $(-3, 4)$ has a slope of -3. What is the value of $j$?
The problem requires finding the unknown coordinate \(j\) of the point \(P_1 = (-4, j)\) on a line that also passes through the point \(P_2 = (-3, 4)\) and has a given slope \(m = -3\). The slope of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is defined by the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\). We have \((x_1, y_1) = (-4, j)\) and \((x_2, y_2) = (-3, 4)\).
Using the slope formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\), we substitute our known values:
\[-3 = \frac{4 - j}{-3 - (-4)}\]
Simplifying the denominator:
\[-3 - (-4) = -3 + 4 = 1\]
So, the equation becomes:
\[-3 = \frac{4 - j}{1}\]
\[-3 = 4 - j\]
To isolate \(j\), we rearrange the equation \(4 - j = -3\):
\[-j = -3 - 4\]
\[-j = -7\]
Multiplying both sides by \(-1\), we find:
\[j = 7\]
Thus, the value of \(j\) is 7, and the line passes through the point \((-4, 7)\).
Slope is a fundamental concept in engineering and design, representing the rate of change between two variables. For example, if you are designing a wheelchair ramp, the slope (or grade) determines how steep the ramp is relative to its horizontal distance. By knowing the required rise and the allowed slope, you can calculate the necessary horizontal reach to ensure safety and accessibility, just as we calculated the coordinate on a line given its slope.