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Sagot :
Sure, let's go through the steps to graph the line with slope [tex]\(\frac{3}{4}\)[/tex] passing through the point [tex]\((-5,-5)\)[/tex].
### Step-by-Step Solution:
1. Understanding the Line Equation:
The general form of the equation of a line is
[tex]\[ y = mx + b \][/tex]
where [tex]\(m\)[/tex] is the slope and [tex]\(b\)[/tex] is the y-intercept.
2. Given Information:
- Slope ([tex]\(m\)[/tex]) = [tex]\(\frac{3}{4}\)[/tex]
- Point ([tex]\(x_1, y_1\)[/tex]) = [tex]\((-5, -5)\)[/tex]
3. Find the y-Intercept:
To find the [tex]\(y\)[/tex]-intercept ([tex]\(b\)[/tex]), we will use the point-slope form of the equation. The point-slope form of the equation is given by:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Substituting the values [tex]\(m = \frac{3}{4}\)[/tex], [tex]\(x_1 = -5\)[/tex], and [tex]\(y_1 = -5\)[/tex], we get:
[tex]\[ y - (-5) = \frac{3}{4}(x - (-5)) \][/tex]
Simplifying this:
[tex]\[ y + 5 = \frac{3}{4}(x + 5) \][/tex]
To convert this into the slope-intercept form [tex]\(y = mx + b\)[/tex], solve for [tex]\(y\)[/tex]:
[tex]\[ y + 5 = \frac{3}{4}x + \frac{3}{4} \cdot 5 \][/tex]
[tex]\[ y + 5 = \frac{3}{4}x + \frac{15}{4} \][/tex]
[tex]\[ y = \frac{3}{4}x + \frac{15}{4} - 5 \][/tex]
[tex]\[ y = \frac{3}{4}x + \frac{15}{4} - \frac{20}{4} \][/tex]
[tex]\[ y = \frac{3}{4}x - \frac{5}{4} \][/tex]
So, the equation of the line is:
[tex]\[ y = \frac{3}{4}x - \frac{5}{4} \][/tex]
4. Plot the Graph:
- Plot the Point: Start by plotting the point [tex]\((-5, -5)\)[/tex] on the coordinate plane.
- Plot the Line:
- Find the y-intercept ([tex]\(b\)[/tex]): [tex]\(\left(0, -\frac{5}{4}\right)\)[/tex].
- Use the slope to find another point: From the y-intercept, move up 3 units and right 4 units to get another point on the line.
- Draw the line passing through these points.
5. Draw the Line:
- Begin at the y-intercept [tex]\(\left(0, -\frac{5}{4}\right)\)[/tex].
- Use the slope [tex]\(\frac{3}{4}\)[/tex] to find additional points (e.g., from [tex]\(0, -\frac{5}{4}\)[/tex], move up 3 and right 4 to [tex]\((4, \frac{7}{4})\)[/tex]).
- Connect these points with a straight line, ensuring it extends across the graph.
### Graph:
You should get a graph that passes through the point [tex]\((-5, -5)\)[/tex] and follows the slope [tex]\(\frac{3}{4}\)[/tex].
Summary:
- The line equation is [tex]\(y = \frac{3}{4}x - \frac{5}{4}\)[/tex].
- Plot the point [tex]\((-5, -5)\)[/tex].
- Use the slope to determine the tilt and direction of the line, plotting points accordingly.
- Draw the line through the plotted points to complete the graph.
### Step-by-Step Solution:
1. Understanding the Line Equation:
The general form of the equation of a line is
[tex]\[ y = mx + b \][/tex]
where [tex]\(m\)[/tex] is the slope and [tex]\(b\)[/tex] is the y-intercept.
2. Given Information:
- Slope ([tex]\(m\)[/tex]) = [tex]\(\frac{3}{4}\)[/tex]
- Point ([tex]\(x_1, y_1\)[/tex]) = [tex]\((-5, -5)\)[/tex]
3. Find the y-Intercept:
To find the [tex]\(y\)[/tex]-intercept ([tex]\(b\)[/tex]), we will use the point-slope form of the equation. The point-slope form of the equation is given by:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Substituting the values [tex]\(m = \frac{3}{4}\)[/tex], [tex]\(x_1 = -5\)[/tex], and [tex]\(y_1 = -5\)[/tex], we get:
[tex]\[ y - (-5) = \frac{3}{4}(x - (-5)) \][/tex]
Simplifying this:
[tex]\[ y + 5 = \frac{3}{4}(x + 5) \][/tex]
To convert this into the slope-intercept form [tex]\(y = mx + b\)[/tex], solve for [tex]\(y\)[/tex]:
[tex]\[ y + 5 = \frac{3}{4}x + \frac{3}{4} \cdot 5 \][/tex]
[tex]\[ y + 5 = \frac{3}{4}x + \frac{15}{4} \][/tex]
[tex]\[ y = \frac{3}{4}x + \frac{15}{4} - 5 \][/tex]
[tex]\[ y = \frac{3}{4}x + \frac{15}{4} - \frac{20}{4} \][/tex]
[tex]\[ y = \frac{3}{4}x - \frac{5}{4} \][/tex]
So, the equation of the line is:
[tex]\[ y = \frac{3}{4}x - \frac{5}{4} \][/tex]
4. Plot the Graph:
- Plot the Point: Start by plotting the point [tex]\((-5, -5)\)[/tex] on the coordinate plane.
- Plot the Line:
- Find the y-intercept ([tex]\(b\)[/tex]): [tex]\(\left(0, -\frac{5}{4}\right)\)[/tex].
- Use the slope to find another point: From the y-intercept, move up 3 units and right 4 units to get another point on the line.
- Draw the line passing through these points.
5. Draw the Line:
- Begin at the y-intercept [tex]\(\left(0, -\frac{5}{4}\right)\)[/tex].
- Use the slope [tex]\(\frac{3}{4}\)[/tex] to find additional points (e.g., from [tex]\(0, -\frac{5}{4}\)[/tex], move up 3 and right 4 to [tex]\((4, \frac{7}{4})\)[/tex]).
- Connect these points with a straight line, ensuring it extends across the graph.
### Graph:
You should get a graph that passes through the point [tex]\((-5, -5)\)[/tex] and follows the slope [tex]\(\frac{3}{4}\)[/tex].
Summary:
- The line equation is [tex]\(y = \frac{3}{4}x - \frac{5}{4}\)[/tex].
- Plot the point [tex]\((-5, -5)\)[/tex].
- Use the slope to determine the tilt and direction of the line, plotting points accordingly.
- Draw the line through the plotted points to complete the graph.
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