Find an equation for the line that passes through the points (-5, -2) and (-1, 4).

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05-11-2026
Mathematics

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Find an equation for the line that passes through the points (-5, -2) and (-1, 4).

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GinnyAnswer GinnyAnswer · Answer 1 · 2026-11-05T10:34:38-05:00

Determine the slope of the line passing through $(-5, -2)$ and $(-1, 4)$ using the formula $m = \frac{y_2 - y_1}{x_2 - x_1}$, resulting in $m = \frac{3}{2}$.
Use the point-slope form $y - y_1 = m(x - x_1)$ to derive the equation $y - 4 = \frac{3}{2}(x + 1)$.
Simplify to the slope-intercept form: $y = \frac{3}{2}x + \frac{11}{2}$.
Express the final equation as: $\boxed{y = \frac{3}{2}x + \frac{11}{2}}$.

Explanation

Calculate the slope of the line.

To determine the equation of a line passing through two points, we first calculate its slope. Given two points $(x_1, y_1) = (-5, -2)$ and $(x_2, y_2) = (-1, 4)$, the slope $m$ is defined as the ratio of the change in $y$ to the change in $x$:
\[m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{4 - (-2)}{-1 - (-5)} = \frac{6}{4} = \frac{3}{2}\]
The slope of the line is $\frac{3}{2}$.

Find the equation of the line.

Now that we have the slope $m = \frac{3}{2}$, we can use the point-slope form of a linear equation, which is $y - y_1 = m(x - x_1)$. Substituting the point $(-1, 4)$ into this form:
\[y - 4 = \frac{3}{2}(x - (-1))\]
\[y - 4 = \frac{3}{2}(x + 1)\]
\[y - 4 = \frac{3}{2}x + \frac{3}{2}\]
Adding $4$ (or $\frac{8}{2}$) to both sides gives the slope-intercept form:
\[y = \frac{3}{2}x + \frac{3}{2} + \frac{8}{2}\]
\[y = \frac{3}{2}x + \frac{11}{2}\]
This is the equation of the line in slope-intercept form.

Verify and express the final equation.

We can rewrite the equation in standard form $Ax + By = C$ by multiplying by 2 and rearranging:
\[2y = 3x + 11\]
\[-3x + 2y = 11\]
\[3x - 2y = -11\]
To verify, check the points $(-5, -2)$ and $(-1, 4)$:

For $(-5, -2)$: $3(-5) - 2(-2) = -15 + 4 = -11$ (Matches)
For $(-1, 4)$: $3(-1) - 2(4) = -3 - 8 = -11$ (Matches)

The final equation of the line is $\boxed{y = \frac{3}{2}x + \frac{11}{2}}$ or $\boxed{3x - 2y = -11}$.

Examples

Understanding how to find the equation of a line is essential in fields like engineering and economics. For example, if you are tracking the linear growth of a project's cost relative to time, you can define two points $(time_1, cost_1)$ and $(time_2, cost_2)$ and use the linear equation to predict future costs. If your project starts at $\$2000$ and grows at a steady rate, finding this line helps in budgeting for future months. Calculations like $m = \frac{y_2 - y_1}{x_2 - x_1}$ become powerful tools for planning and forecasting.