To graph a line with a given slope and a point that it passes through, we need to determine the equation of the line in the slope-intercept form, which is [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept. Here’s a step-by-step process:
1. Identify the given slope and point:
- The slope ([tex]\( m \)[/tex]) is -5.
- The point is [tex]\((-3, -4)\)[/tex], where [tex]\( x = -3 \)[/tex] and [tex]\( y = -4 \)[/tex].
2. Substitute the given point and the slope into the slope-intercept formula:
We use the point [tex]\((-3, -4)\)[/tex] to find the y-intercept ([tex]\( b \)[/tex]).
[tex]\[
y = mx + b
\][/tex]
Substitute [tex]\( y = -4 \)[/tex], [tex]\( x = -3 \)[/tex], and [tex]\( m = -5 \)[/tex]:
[tex]\[
-4 = -5(-3) + b
\][/tex]
3. Solve for the y-intercept ([tex]\( b \)[/tex]):
[tex]\[
-4 = 15 + b
\][/tex]
Subtract 15 from both sides to isolate [tex]\( b \)[/tex]:
[tex]\[
b = -4 - 15
\][/tex]
[tex]\[
b = -19
\][/tex]
4. Write the equation of the line:
Now that we have the y-intercept ([tex]\( b = -19 \)[/tex]), we can write the equation with the given slope ([tex]\( m = -5 \)[/tex]):
[tex]\[
y = -5x - 19
\][/tex]
5. Graph the line:
- Start by plotting the y-intercept [tex]\((0, -19)\)[/tex] on the coordinate plane.
- From the y-intercept, use the slope to find another point on the line. The slope of -5 means that for every 1 unit you move to the right (positive direction on the x-axis), you move 5 units down (negative direction on the y-axis).
- From [tex]\((0, -19)\)[/tex], move 1 unit to the right to [tex]\((1, -19)\)[/tex], then 5 units down to [tex]\((1, -24)\)[/tex]. Plot this point [tex]\((1, -24)\)[/tex].
- Draw a straight line through these points, extending it in both directions.
By following these steps, you will have successfully graphed the line that has a slope of -5 and contains the point [tex]\((-3, -4)\)[/tex] with the equation [tex]\( y = -5x - 19 \)[/tex].
