Use Point-slope Form To Write The Equation Of A Line That Passes Through The Point \[$(-11, -8)\$\] With Slope \[$-\frac{3}{2}\$\].Answer: \[$\square\$\]
Introduction
In mathematics, the point-slope form is a method used to write the equation of a line that passes through a given point and has a specified slope. This form is particularly useful when we know the coordinates of a point on the line and the slope of the line. In this article, we will explore how to use the point-slope form to write the equation of a line that passes through the point (-11, -8) with a slope of -3/2.
What is Point-Slope Form?
The point-slope form of a linear equation is given by the formula:
y - y1 = m(x - x1)
where (x1, y1) is the given point on the line, and m is the slope of the line. This formula allows us to write the equation of a line that passes through a specific point and has a specified slope.
Step 1: Identify the Given Point and Slope
In this problem, we are given the point (-11, -8) and the slope -3/2. We will use these values to write the equation of the line.
Step 2: Plug in the Values into the Point-Slope Formula
Now that we have identified the given point and slope, we can plug these values into the point-slope formula:
y - y1 = m(x - x1)
Substituting the given values, we get:
y - (-8) = -3/2(x - (-11))
Step 3: Simplify the Equation
To simplify the equation, we can start by evaluating the expressions inside the parentheses:
y + 8 = -3/2(x + 11)
Next, we can distribute the slope to the terms inside the parentheses:
y + 8 = -3/2x - 33/2
Step 4: Write the Equation in Standard Form
To write the equation in standard form, we can add 33/2 to both sides of the equation:
y + 8 + 33/2 = -3/2x
Combining the constant terms on the left-hand side, we get:
y + 65/2 = -3/2x
Step 5: Write the Equation in Slope-Intercept Form
To write the equation in slope-intercept form, we can subtract 65/2 from both sides of the equation:
y = -3/2x - 65/2
Conclusion
In this article, we used the point-slope form to write the equation of a line that passes through the point (-11, -8) with a slope of -3/2. We started by identifying the given point and slope, then plugged these values into the point-slope formula. We simplified the equation and wrote it in standard form, and finally, we wrote it in slope-intercept form. This process demonstrates how to use the point-slope form to find the equation of a line that passes through a given point and has a specified slope.
Example Problems
- Write the equation of a line that passes through the point (2, 3) with a slope of 2/3.
- Write the equation of a line that passes through the point (-4, 2) with a slope of -1/2.
- Write the equation of a line that passes through the point (1, -2) with a slope of 3/4.
Practice Problems
- Write the equation of a line that passes through the point (-5, 1) with a slope of -2/3.
- Write the equation of a line that passes through the point (3, -4) with a slope of 1/2.
- Write the equation of a line that passes through the point (-2, 5) with a slope of -3/4.
Glossary of Terms
- Point-slope form: A method used to write the equation of a line that passes through a given point and has a specified slope.
- Slope: A measure of the steepness of a line, calculated as the ratio of the vertical change to the horizontal change.
- Standard form: A way of writing the equation of a line, with the x-term on the left-hand side and the constant term on the right-hand side.
- Slope-intercept form: A way of writing the equation of a line, with the slope as the coefficient of the x-term and the y-intercept as the constant term.
Frequently Asked Questions: Point-Slope Form =====================================================
Q: What is the point-slope form of a linear equation?
A: The point-slope form of a linear equation is given by the formula:
y - y1 = m(x - x1)
where (x1, y1) is the given point on the line, and m is the slope of the line.
Q: How do I use the point-slope form to write the equation of a line?
A: To use the point-slope form, you need to identify the given point and slope, then plug these values into the formula. Simplify the equation and write it in standard form, and finally, write it in slope-intercept form.
Q: What is the difference between the point-slope form and the slope-intercept form?
A: The point-slope form is used to write the equation of a line that passes through a given point and has a specified slope. The slope-intercept form is used to write the equation of a line in the form y = mx + b, where m is the slope and b is the y-intercept.
Q: Can I use the point-slope form to write the equation of a line that is not in the standard form?
A: Yes, you can use the point-slope form to write the equation of a line that is not in the standard form. However, you need to first rewrite the equation in the standard form before using the point-slope form.
Q: How do I find the slope of a line given two points?
A: To find the slope of a line given two points, you can use the formula:
m = (y2 - y1) / (x2 - x1)
where (x1, y1) and (x2, y2) are the two points on the line.
Q: Can I use the point-slope form to write the equation of a line that is vertical?
A: No, you cannot use the point-slope form to write the equation of a line that is vertical. The point-slope form is used to write the equation of a line that has a non-zero slope.
Q: How do I determine if a line is vertical or horizontal?
A: To determine if a line is vertical or horizontal, you can look at the slope of the line. If the slope is zero, the line is horizontal. If the slope is undefined, the line is vertical.
Q: Can I use the point-slope form to write the equation of a line that is horizontal?
A: Yes, you can use the point-slope form to write the equation of a line that is horizontal. However, you need to use the formula:
y - y1 = 0(x - x1)
where (x1, y1) is the given point on the line.
Q: How do I find the equation of a line that passes through a given point and has a specified slope?
A: To find the equation of a line that passes through a given point and has a specified slope, you can use the point-slope form. Plug in the given values into the formula, simplify the equation, and write it in standard form and slope-intercept form.
Q: Can I use the point-slope form to write the equation of a line that is a circle?
A: No, you cannot use the point-slope form to write the equation of a line that is a circle. The point-slope form is used to write the equation of a line that has a non-zero slope. A circle has a zero slope.
Q: How do I determine if a line is a circle or not?
A: To determine if a line is a circle or not, you can look at the equation of the line. If the equation is in the form (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center of the circle and r is the radius, then the line is a circle. If the equation is in the form y = mx + b, where m is the slope and b is the y-intercept, then the line is not a circle.
Q: Can I use the point-slope form to write the equation of a line that is a parabola?
A: No, you cannot use the point-slope form to write the equation of a line that is a parabola. The point-slope form is used to write the equation of a line that has a non-zero slope. A parabola has a zero slope.
Q: How do I determine if a line is a parabola or not?
A: To determine if a line is a parabola or not, you can look at the equation of the line. If the equation is in the form y = ax^2 + bx + c, where a, b, and c are constants, then the line is a parabola. If the equation is in the form y = mx + b, where m is the slope and b is the y-intercept, then the line is not a parabola.