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Introduction
In mathematics, linear equations are a fundamental concept that describe a relationship between two or more variables. The point-slope form and slope-intercept form are two common ways to represent a linear equation. In this article, we will explore how to write an equation for a line in point-slope form and slope-intercept form, given a point on the line and its x-intercept.
Point-Slope Form
The point-slope form of a linear equation is given by:
y - y1 = m(x - x1)
where (x1, y1) is a point on the line, and m is the slope of the line.
Given Conditions
We are given a point on the line (4, 9) and its x-intercept (-1). To write an equation for the line in point-slope form, we need to find the slope of the line.
Finding the Slope
The slope of a line can be found using the formula:
m = (y2 - y1) / (x2 - x1)
where (x1, y1) and (x2, y2) are two points on the line.
In this case, we can use the given point (4, 9) and the x-intercept (-1, 0) to find the slope:
m = (9 - 0) / (4 - (-1)) m = 9 / 5 m = 1.8
Writing the Equation in Point-Slope Form
Now that we have the slope, we can write the equation for the line in point-slope form:
y - 9 = 1.8(x - 4)
Simplifying the Equation
To simplify the equation, we can expand the right-hand side:
y - 9 = 1.8x - 7.2
Adding 9 to Both Sides
To isolate y, we can add 9 to both sides of the equation:
y = 1.8x - 7.2 + 9 y = 1.8x + 1.8
Slope-Intercept Form
The slope-intercept form of a linear equation is given by:
y = mx + b
where m is the slope of the line, and b is the y-intercept.
Finding the Y-Intercept
To find the y-intercept, we can substitute x = 0 into the equation:
y = 1.8(0) + 1.8 y = 1.8
Writing the Equation in Slope-Intercept Form
Now that we have the slope and y-intercept, we can write the equation for the line in slope-intercept form:
y = 1.8x + 1.8
Conclusion
In this article, we have seen how to write an equation for a line in point-slope form and slope-intercept form, given a point on the line and its x-intercept. We have also seen how to find the slope and y-intercept of a line, and how to simplify the equation in point-slope form.
Key Takeaways
- The point-slope form of a linear equation is given by y - y1 = m(x - x1).
- The slope of a line can be found using the formula m = (y2 - y1) / (x2 - x1).
- The slope-intercept form of a linear equation is given by y = mx + b.
- The y-intercept of a line can be found by substituting x = 0 into the equation.
Practice Problems
- Write an equation for the line in point-slope form, given the point (2, 3) and the x-intercept (-1).
- Write an equation for the line in slope-intercept form, given the point (4, 9) and the slope 2.
- Find the slope and y-intercept of the line y = 2x + 3.
- Write an equation for the line in point-slope form, given the point (1, 2) and the slope 3.
- Write an equation for the line in slope-intercept form, given the point (3, 4) and the y-intercept 2.
Answers
- y - 3 = 2(x - 2)
- y = 2x + 9
- m = 2, b = 3
- y - 2 = 3(x - 1)
- y = 2x + 2
Frequently Asked Questions: Point-Slope and Slope-Intercept Forms ====================================================================
Q: What is the point-slope form of a linear equation?
A: The point-slope form of a linear equation is given by:
y - y1 = m(x - x1)
where (x1, y1) is a point on the line, and m is the slope of the line.
Q: How do I find the slope of a line?
A: The slope of a line can be found using the formula:
m = (y2 - y1) / (x2 - x1)
where (x1, y1) and (x2, y2) are two points on the line.
Q: What is the slope-intercept form of a linear equation?
A: The slope-intercept form of a linear equation is given by:
y = mx + b
where m is the slope of the line, and b is the y-intercept.
Q: How do I find the y-intercept of a line?
A: The y-intercept of a line can be found by substituting x = 0 into the equation.
Q: Can I use the point-slope form to find the slope of a line?
A: Yes, you can use the point-slope form to find the slope of a line. Simply substitute the given point and the x-intercept into the equation, and solve for the slope.
Q: Can I use the slope-intercept form to find the y-intercept of a line?
A: Yes, you can use the slope-intercept form to find the y-intercept of a line. Simply substitute x = 0 into the equation, and solve for the y-intercept.
Q: How do I write an equation for a line in point-slope form?
A: To write an equation for a line in point-slope form, you need to know the slope of the line and a point on the line. Use the point-slope form equation:
y - y1 = m(x - x1)
and substitute the given point and the slope into the equation.
Q: How do I write an equation for a line in slope-intercept form?
A: To write an equation for a line in slope-intercept form, you need to know the slope of the line and the y-intercept of the line. Use the slope-intercept form equation:
y = mx + b
and substitute the given slope and y-intercept into the equation.
Q: Can I use the point-slope form and slope-intercept form to solve a system of linear equations?
A: Yes, you can use the point-slope form and slope-intercept form to solve a system of linear equations. Simply write the equations in both forms, and then solve the system using either the point-slope form or the slope-intercept form.
Q: What are some common mistakes to avoid when working with point-slope and slope-intercept forms?
A: Some common mistakes to avoid when working with point-slope and slope-intercept forms include:
- Not using the correct formula for the slope
- Not substituting the correct values into the equation
- Not simplifying the equation correctly
- Not checking the solution for extraneous solutions
Q: How do I check my solution for extraneous solutions?
A: To check your solution for extraneous solutions, simply substitute the solution back into the original equation and check if it is true. If it is not true, then the solution is extraneous and should be discarded.
Q: Can I use the point-slope form and slope-intercept form to graph a line?
A: Yes, you can use the point-slope form and slope-intercept form to graph a line. Simply plot the given point and the x-intercept, and then use the slope to draw the line.
Q: What are some real-world applications of point-slope and slope-intercept forms?
A: Some real-world applications of point-slope and slope-intercept forms include:
- Finding the equation of a line that passes through a given point and has a given slope
- Finding the equation of a line that passes through two given points
- Finding the equation of a line that has a given slope and y-intercept
- Graphing a line using the point-slope form and slope-intercept form
Conclusion
In this article, we have seen how to use the point-slope form and slope-intercept form to solve linear equations and graph lines. We have also seen some common mistakes to avoid and some real-world applications of these forms.