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To graph the line with a slope of [tex]\(-\frac{3}{4}\)[/tex] that passes through the point [tex]\((2, 3)\)[/tex], follow these steps:
1. Understand the Slope-Intercept Form:
The slope-intercept form of a linear equation is given by:
[tex]\[ y = mx + b \][/tex]
where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
2. Identify the Provided Information:
- Slope ([tex]\(m\)[/tex]) is given as [tex]\(-\frac{3}{4}\)[/tex].
- The line passes through the point [tex]\((2, 3)\)[/tex]. Here, [tex]\(x_1 = 2\)[/tex] and [tex]\(y_1 = 3\)[/tex].
3. Substitute the Slope and Point into the Point-Slope Form:
The point-slope form of a line's equation is:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Substituting [tex]\(m = -\frac{3}{4}\)[/tex], [tex]\(x_1 = 2\)[/tex], and [tex]\(y_1 = 3\)[/tex]:
[tex]\[ y - 3 = -\frac{3}{4}(x - 2) \][/tex]
4. Convert to Slope-Intercept Form:
Simplify the equation to get it in the form [tex]\(y = mx + b\)[/tex]:
[tex]\[ y - 3 = -\frac{3}{4}x + \frac{3}{2} \][/tex]
Add 3 to both sides to solve for [tex]\(y\)[/tex]:
[tex]\[ y = -\frac{3}{4}x + \frac{3}{2} + 3 \][/tex]
Express 3 as a fraction with a common denominator:
[tex]\[ y = -\frac{3}{4}x + \frac{3}{2} + \frac{6}{2} \][/tex]
Combine the fractions:
[tex]\[ y = -\frac{3}{4}x + \frac{9}{2} \][/tex]
Therefore:
[tex]\[ b = \frac{9}{2} \][/tex]
Converting [tex]\(\frac{9}{2}\)[/tex] to a decimal for clarity:
[tex]\[ b = 4.5 \][/tex]
5. Write the Final Equation:
The equation of the line in slope-intercept form is:
[tex]\[ y = -\frac{3}{4}x + 4.5 \][/tex]
6. Plot the Line:
- Start by plotting the y-intercept ([tex]\(0, 4.5\)[/tex]).
- From this point, use the slope [tex]\(-\frac{3}{4}\)[/tex]:
- Move down 3 units (because of the negative slope).
- Move right 4 units.
- Alternatively, you can use the given point [tex]\((2, 3)\)[/tex]. Start at [tex]\((2, 3)\)[/tex] and apply the slope:
- From [tex]\((2, 3)\)[/tex], moving down 3 units and right 4 units should maintain the line.
7. Draw the Line:
Connect these points with a straight line, ensuring the plotted points align with the slope of [tex]\(-\frac{3}{4}\)[/tex].
The graph of this line will show a descending slope from left to right, intersecting the y-axis at [tex]\(4.5\)[/tex] and passing through the point [tex]\((2, 3)\)[/tex].
1. Understand the Slope-Intercept Form:
The slope-intercept form of a linear equation is given by:
[tex]\[ y = mx + b \][/tex]
where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
2. Identify the Provided Information:
- Slope ([tex]\(m\)[/tex]) is given as [tex]\(-\frac{3}{4}\)[/tex].
- The line passes through the point [tex]\((2, 3)\)[/tex]. Here, [tex]\(x_1 = 2\)[/tex] and [tex]\(y_1 = 3\)[/tex].
3. Substitute the Slope and Point into the Point-Slope Form:
The point-slope form of a line's equation is:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Substituting [tex]\(m = -\frac{3}{4}\)[/tex], [tex]\(x_1 = 2\)[/tex], and [tex]\(y_1 = 3\)[/tex]:
[tex]\[ y - 3 = -\frac{3}{4}(x - 2) \][/tex]
4. Convert to Slope-Intercept Form:
Simplify the equation to get it in the form [tex]\(y = mx + b\)[/tex]:
[tex]\[ y - 3 = -\frac{3}{4}x + \frac{3}{2} \][/tex]
Add 3 to both sides to solve for [tex]\(y\)[/tex]:
[tex]\[ y = -\frac{3}{4}x + \frac{3}{2} + 3 \][/tex]
Express 3 as a fraction with a common denominator:
[tex]\[ y = -\frac{3}{4}x + \frac{3}{2} + \frac{6}{2} \][/tex]
Combine the fractions:
[tex]\[ y = -\frac{3}{4}x + \frac{9}{2} \][/tex]
Therefore:
[tex]\[ b = \frac{9}{2} \][/tex]
Converting [tex]\(\frac{9}{2}\)[/tex] to a decimal for clarity:
[tex]\[ b = 4.5 \][/tex]
5. Write the Final Equation:
The equation of the line in slope-intercept form is:
[tex]\[ y = -\frac{3}{4}x + 4.5 \][/tex]
6. Plot the Line:
- Start by plotting the y-intercept ([tex]\(0, 4.5\)[/tex]).
- From this point, use the slope [tex]\(-\frac{3}{4}\)[/tex]:
- Move down 3 units (because of the negative slope).
- Move right 4 units.
- Alternatively, you can use the given point [tex]\((2, 3)\)[/tex]. Start at [tex]\((2, 3)\)[/tex] and apply the slope:
- From [tex]\((2, 3)\)[/tex], moving down 3 units and right 4 units should maintain the line.
7. Draw the Line:
Connect these points with a straight line, ensuring the plotted points align with the slope of [tex]\(-\frac{3}{4}\)[/tex].
The graph of this line will show a descending slope from left to right, intersecting the y-axis at [tex]\(4.5\)[/tex] and passing through the point [tex]\((2, 3)\)[/tex].
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