To graph a line with a slope of [tex]\(-5\)[/tex] that passes through the point [tex]\((-3, -4)\)[/tex], we need to follow these steps:
1. Identify the slope-intercept form:
The slope-intercept form of a line is given by:
[tex]\[
y = mx + b
\][/tex]
where [tex]\(m\)[/tex] is the slope and [tex]\(b\)[/tex] is the y-intercept.
2. Use the given point to find the y-intercept:
Given the point [tex]\((-3, -4)\)[/tex] and the slope [tex]\(m = -5\)[/tex], we substitute these values into the slope-intercept form to find [tex]\(b\)[/tex].
Substitute [tex]\(x = -3\)[/tex], [tex]\(y = -4\)[/tex], and [tex]\(m = -5\)[/tex] into the equation [tex]\(y = mx + b\)[/tex]:
[tex]\[
-4 = (-5)(-3) + b
\][/tex]
Calculate the right-hand side:
[tex]\[
-4 = 15 + b
\][/tex]
Solving for [tex]\(b\)[/tex]:
[tex]\[
b = -4 - 15 = -19
\][/tex]
Thus, the equation of the line is:
[tex]\[
y = -5x - 19
\][/tex]
3. Choose additional points for the graph:
To graph the line, it’s helpful to find a couple more points. We can choose any values for [tex]\(x\)[/tex] and then calculate the corresponding [tex]\(y\)[/tex] values using the equation [tex]\(y = -5x - 19\)[/tex].
- For [tex]\(x = 0\)[/tex]:
[tex]\[
y = -5(0) - 19 = -19
\][/tex]
So, one point is [tex]\((0, -19)\)[/tex].
- For [tex]\(x = 2\)[/tex]:
[tex]\[
y = -5(2) - 19 = -10 - 19 = -29
\][/tex]
So, another point is [tex]\((2, -29)\)[/tex].
4. Plot the points and draw the line:
- Plot the point [tex]\((-3, -4)\)[/tex].
- Plot the point [tex]\((0, -19)\)[/tex].
- Plot the point [tex]\((2, -29)\)[/tex].
- Draw a straight line through these points.
Here's a sketch of how the plot would look:
[tex]\[
\begin{array}{c|c}
x & y \\
\hline
-3 & -4 \\
0 & -19 \\
2 & -29 \\
\end{array}
\][/tex]
Note: The y-values are negative, so you'll need to accommodate for that in your graph. Your y-axis should extend far enough to capture the values like -19 and -29.
Thus, by plotting these points and connecting them, you will have the graph of the line with a slope of [tex]\(-5\)[/tex] that contains the point [tex]\((-3, -4)\)[/tex].
