A line has a slope of $-\frac{9}{4}$ and includes the points (-7, 1) and (-3, d). What is the value of $d$?

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06-11-2026
Mathematics

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A line has a slope of $-\frac{9}{4}$ and includes the points (-7, 1) and (-3, d). What is the value of $d$?

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GinnyAnswer GinnyAnswer · Answer 1 · 2026-11-06T14:37:20-05:00

The slope formula is defined as $m = \frac{y_2 - y_1}{x_2 - x_1}$.
Substituting the given slope $m = -\frac{9}{4}$ and points $(-7, 1)$ and $(-3, d)$ into the formula yields $-\frac{9}{4} = \frac{d-1}{4}$.
Multiplying both sides by 4 and simplifying gives $d-1 = -9$.
Solving for $d$ results in $d = -8$, so $\boxed{-8}$.

Explanation

Analysis of the Problem

To find the value of $d$, we use the definition of the slope of a line passing through two points. We are given that the line has a slope $m = -\frac{9}{4}$ and passes through the points $P_1 = (-7, 1)$ and $P_2 = (-3, d)$. The slope $m$ between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by the formula:
\[m = \frac{y_2 - y_1}{x_2 - x_1}\]
Our goal is to substitute the given values into this formula and solve for the unknown coordinate $d$.

Setting up the Equation

Using the coordinates of the two points $(-7, 1)$ and $(-3, d)$, we identify $x_1 = -7, y_1 = 1, x_2 = -3, y_2 = d$. Plugging these into the slope formula, we get:
\[-\frac{9}{4} = \frac{d - 1}{-3 - (-7)}\]
We simplify the denominator:
\[-3 - (-7) = -3 + 7 = 4\]
So the equation becomes:
\[-\frac{9}{4} = \frac{d - 1}{4}\]

Solving for d

Now, we solve for $d$. Multiplying both sides of the equation by 4, we eliminate the denominators:
\[4 \times \left(-\frac{9}{4}\right) = 4 \times \left(\frac{d - 1}{4}\right)\]
\[-9 = d - 1\]
Adding 1 to both sides:
\[d = -9 + 1\]
\[d = -8\]
Thus, the value of $d$ is $-8$.

Examples

In navigation and civil engineering, understanding slopes is crucial for designing roads or ramps. If a engineer needs to ensure a ramp drops by a specific amount over a certain distance to comply with safety standards, they use the slope formula. For example, if a design requires a downward slope, knowing one point on the path and the required steepness allows the engineer to determine the exact vertical position needed at any given horizontal distance, ensuring accessibility and structural integrity.