Discover the best answers at Westonci.ca, where experts share their insights and knowledge with you. Get accurate and detailed answers to your questions from a dedicated community of experts on our Q&A platform. Discover in-depth answers to your questions from a wide network of professionals on our user-friendly Q&A platform.
Sagot :
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].
We hope our answers were helpful. Return anytime for more information and answers to any other questions you may have. Thank you for visiting. Our goal is to provide the most accurate answers for all your informational needs. Come back soon. Westonci.ca is here to provide the answers you seek. Return often for more expert solutions.
