IDNLearn.com: Your trusted source for finding accurate answers. Whether it's a simple query or a complex problem, our experts have the answers you need.
Sagot :
Sure, let's graph the line with slope [tex]\( -\frac{1}{3} \)[/tex] passing through the point [tex]\( (5, 4) \)[/tex]. Here is a step-by-step solution for how to do this:
1. Identify the Point-Slope Formula:
The point-slope formula of a line is given by:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
where [tex]\( (x_1, y_1) \)[/tex] is a point on the line and [tex]\( m \)[/tex] is the slope.
2. Substitute the Given Values:
We know the slope [tex]\( m = -\frac{1}{3} \)[/tex], and the point [tex]\( (x_1, y_1) = (5, 4) \)[/tex]. Substituting these values into the equation, we get:
[tex]\[ y - 4 = -\frac{1}{3}(x - 5) \][/tex]
3. Simplify the Equation:
Distribute [tex]\( -\frac{1}{3} \)[/tex] on the right-hand side:
[tex]\[ y - 4 = -\frac{1}{3}x + \frac{5}{3} \][/tex]
Add 4 to both sides to solve for [tex]\( y \)[/tex]:
[tex]\[ y = -\frac{1}{3}x + \frac{5}{3} + 4 \][/tex]
Convert 4 to a fraction with a common denominator:
[tex]\[ y = -\frac{1}{3}x + \frac{5}{3} + \frac{12}{3} \][/tex]
Combine the fractions:
[tex]\[ y = -\frac{1}{3}x + \frac{17}{3} \][/tex]
4. Plot the Line:
- Start by plotting the point [tex]\( (5, 4) \)[/tex] on the graph.
- Use the slope to find other points. Since the slope is [tex]\( -\frac{1}{3} \)[/tex], this means that for every 3 units you move horizontally to the right, you move 1 unit vertically downwards.
Let's find another point using the slope:
- From [tex]\( (5, 4) \)[/tex], move 3 units to the right to [tex]\( x = 8 \)[/tex].
- Move 1 unit down to [tex]\( y = 3 \)[/tex].
- So, another point on the line is [tex]\( (8, 3) \)[/tex].
5. Draw the Line:
- Plot the point [tex]\( (8, 3) \)[/tex] on the graph.
- Draw a straight line passing through the two points [tex]\( (5, 4) \)[/tex] and [tex]\( (8, 3) \)[/tex].
6. Label the Graph:
- Mark the points [tex]\( (5, 4) \)[/tex] and [tex]\( (8, 3) \)[/tex] clearly.
- Write the equation of the line [tex]\( y = -\frac{1}{3}x + \frac{17}{3} \)[/tex] on the graph.
- Optionally, draw and label the x and y-axes accurately.
By following these steps, you should have a properly graph of the line passing through the point [tex]\( (5, 4) \)[/tex] with a slope of [tex]\( -\frac{1}{3} \)[/tex].
1. Identify the Point-Slope Formula:
The point-slope formula of a line is given by:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
where [tex]\( (x_1, y_1) \)[/tex] is a point on the line and [tex]\( m \)[/tex] is the slope.
2. Substitute the Given Values:
We know the slope [tex]\( m = -\frac{1}{3} \)[/tex], and the point [tex]\( (x_1, y_1) = (5, 4) \)[/tex]. Substituting these values into the equation, we get:
[tex]\[ y - 4 = -\frac{1}{3}(x - 5) \][/tex]
3. Simplify the Equation:
Distribute [tex]\( -\frac{1}{3} \)[/tex] on the right-hand side:
[tex]\[ y - 4 = -\frac{1}{3}x + \frac{5}{3} \][/tex]
Add 4 to both sides to solve for [tex]\( y \)[/tex]:
[tex]\[ y = -\frac{1}{3}x + \frac{5}{3} + 4 \][/tex]
Convert 4 to a fraction with a common denominator:
[tex]\[ y = -\frac{1}{3}x + \frac{5}{3} + \frac{12}{3} \][/tex]
Combine the fractions:
[tex]\[ y = -\frac{1}{3}x + \frac{17}{3} \][/tex]
4. Plot the Line:
- Start by plotting the point [tex]\( (5, 4) \)[/tex] on the graph.
- Use the slope to find other points. Since the slope is [tex]\( -\frac{1}{3} \)[/tex], this means that for every 3 units you move horizontally to the right, you move 1 unit vertically downwards.
Let's find another point using the slope:
- From [tex]\( (5, 4) \)[/tex], move 3 units to the right to [tex]\( x = 8 \)[/tex].
- Move 1 unit down to [tex]\( y = 3 \)[/tex].
- So, another point on the line is [tex]\( (8, 3) \)[/tex].
5. Draw the Line:
- Plot the point [tex]\( (8, 3) \)[/tex] on the graph.
- Draw a straight line passing through the two points [tex]\( (5, 4) \)[/tex] and [tex]\( (8, 3) \)[/tex].
6. Label the Graph:
- Mark the points [tex]\( (5, 4) \)[/tex] and [tex]\( (8, 3) \)[/tex] clearly.
- Write the equation of the line [tex]\( y = -\frac{1}{3}x + \frac{17}{3} \)[/tex] on the graph.
- Optionally, draw and label the x and y-axes accurately.
By following these steps, you should have a properly graph of the line passing through the point [tex]\( (5, 4) \)[/tex] with a slope of [tex]\( -\frac{1}{3} \)[/tex].
We value your participation in this forum. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. For trustworthy answers, visit IDNLearn.com. Thank you for your visit, and see you next time for more reliable solutions.
