Find an equation of a line with a slope of -1 and having the point $(-2,-6)$ on the line. Then graph it.

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Malomtz Malomtz

04-11-2026
Mathematics

Answered

Find an equation of a line with a slope of -1 and having the point $(-2,-6)$ on the line. Then graph it.

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GinnyAnswer GinnyAnswer · Answer 1 · 2026-11-04T03:28:44-05:00

Start with the point-slope formula $y - y_1 = m(x - x_1)$ using $m = -1$ and $(-2, -6)$.
Substitute the values to get $y + 6 = -1(x + 2)$.
Simplify to slope-intercept form to find the final equation: $y = -x - 8$.
The graph is a line passing through $(-2, -6)$, $(0, -8)$, and $(-8, 0)$.
Final Answer: $\boxed{y = -x - 8}$

Explanation

Formulating the Linear Equation

To find the equation of a line, we identify its given properties: the slope $m = -1$ and a point $P = (-2, -6)$ through which it passes. The point-slope form of a linear equation is a powerful tool here:
\[y - y_1 = m(x - x_1)\]
where $(x_1, y_1)$ is the given point. Substituting the known values, we have:
\[y - (-6) = -1(x - (-2))\]
\[y + 6 = -1(x + 2)\]

Converting to Slope-Intercept Form

To simplify the equation into the slope-intercept form, $y = mx + b$, we solve for $y$:
\[y + 6 = -x - 2\]
Subtracting $6$ from both sides:
\[y = -x - 2 - 6\]
\[y = -x - 8\]
This is the equation of the line. The slope is $-1$ and the y-intercept is $-8$.

Graphing the Line

To visualize the line, we identify a few key points:

The given point: $(-2, -6)$
The y-intercept: Setting $x = 0$, we get $y = -0 - 8 = -8$. So, the point is $(0, -8)$.
The x-intercept: Setting $y = 0$, we get $0 = -x - 8$, so $x = -8$. The point is $(-8, 0)$.

By plotting these points on a coordinate plane and drawing a straight line through them, we represent the linear relationship defined by $y = -x - 8$.

Examples

This linear relationship can model real-world scenarios where something decreases at a constant rate. For example, if you have a balance of \$8 in a savings account and withdraw \$1 every day, your balance $y$ after $x$ days could be represented by the equation $y = -1x + 8$. In our case, the negative slope indicates a steady decline over time. Understanding such relationships helps in budget planning, tracking consumption, or predicting how resources deplete over time.