Find The Point-slope Form Of A Line With A Slope Of 2 That Passes Through The Origin.Point-slope Form: \[$(y - \square) = \square(x - \square)\$\]Write The Line In Slope-intercept Form Below:Slope-intercept Form: \[$y = \square\$\]

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Introduction

In mathematics, the point-slope form of a line is a way to express the equation of a line using the slope and a point on the line. This form is particularly useful when we know the slope of the line and a point that it passes through. In this article, we will explore how to find the point-slope form of a line with a slope of 2 that passes through the origin.

Point-Slope Form

The point-slope form of a line is given by the equation:

{(y - \square) = \square(x - \square)$}$

where mm is the slope of the line and (x1,y1)(x_1, y_1) is a point on the line. In this case, we are given that the slope of the line is 2, so we can substitute m=2m = 2 into the equation.

Finding the Point-Slope Form of the Line

Since the line passes through the origin, we know that the point (x1,y1)(x_1, y_1) is (0,0)(0, 0). We can substitute these values into the point-slope form of the equation:

{(y - 0) = 2(x - 0)$}$

Simplifying the equation, we get:

{y = 2x$}$

This is the point-slope form of the line.

Slope-Intercept Form

The slope-intercept form of a line is given by the equation:

{y = mx + b$}$

where mm is the slope of the line and bb is the y-intercept. We can rewrite the point-slope form of the equation in slope-intercept form by substituting m=2m = 2 and b=0b = 0:

{y = 2x + 0$}$

Simplifying the equation, we get:

{y = 2x$}$

This is the slope-intercept form of the line.

Conclusion

In this article, we found the point-slope form of a line with a slope of 2 that passes through the origin. We then converted the point-slope form of the equation to slope-intercept form. The point-slope form of the equation is useful when we know the slope of the line and a point that it passes through, while the slope-intercept form of the equation is useful when we know the slope of the line and the y-intercept.

Example Problems

Problem 1

Find the point-slope form of a line with a slope of 3 that passes through the point (2,1)(2, 1).

Solution

We can substitute m=3m = 3 and (x1,y1)=(2,1)(x_1, y_1) = (2, 1) into the point-slope form of the equation:

{(y - 1) = 3(x - 2)$}$

Simplifying the equation, we get:

{y - 1 = 3x - 6$}$

Adding 1 to both sides of the equation, we get:

{y = 3x - 5$}$

This is the point-slope form of the line.

Problem 2

Find the slope-intercept form of a line with a slope of 4 that passes through the point (1,2)(1, 2).

Solution

We can substitute m=4m = 4 and (x1,y1)=(1,2)(x_1, y_1) = (1, 2) into the point-slope form of the equation:

{(y - 2) = 4(x - 1)$}$

Simplifying the equation, we get:

{y - 2 = 4x - 4$}$

Adding 2 to both sides of the equation, we get:

{y = 4x - 2$}$

This is the slope-intercept form of the line.

Key Takeaways

  • The point-slope form of a line is given by the equation (y−□)=â–¡(x−□)(y - \square) = \square(x - \square).
  • The slope-intercept form of a line is given by the equation y=mx+by = mx + b.
  • We can find the point-slope form of a line by substituting the slope and a point on the line into the point-slope form of the equation.
  • We can find the slope-intercept form of a line by substituting the slope and the y-intercept into the slope-intercept form of the equation.

Glossary of Terms

  • Point-slope form: A way to express the equation of a line using the slope and a point on the line.
  • Slope-intercept form: A way to express the equation of a line using the slope and the y-intercept.
  • Slope: A measure of how steep a line is.
  • Y-intercept: The point where a line intersects the y-axis.
    Frequently Asked Questions (FAQs) =====================================

Q: What is the point-slope form of a line?

A: The point-slope form of a line is a way to express the equation of a line using the slope and a point on the line. It is given by the equation (y−□)=□(x−□)(y - \square) = \square(x - \square), where mm is the slope of the line and (x1,y1)(x_1, y_1) is a point on the line.

Q: How do I find the point-slope form of a line?

A: To find the point-slope form of a line, you need to know the slope of the line and a point on the line. You can then substitute these values into the point-slope form of the equation: (y−y1)=m(x−x1)(y - y_1) = m(x - x_1).

Q: What is the slope-intercept form of a line?

A: The slope-intercept form of a line is a way to express the equation of a line using the slope and the y-intercept. It is given by the equation y=mx+by = mx + b, where mm is the slope of the line and bb is the y-intercept.

Q: How do I find the slope-intercept form of a line?

A: To find the slope-intercept form of a line, you need to know the slope of the line and the y-intercept. You can then substitute these values into the slope-intercept form of the equation: y=mx+by = mx + b.

Q: What is the difference between the point-slope form and the slope-intercept form of a line?

A: The point-slope form of a line uses the slope and a point on the line to express the equation of the line, while the slope-intercept form of a line uses the slope and the y-intercept to express the equation of the line.

Q: Can I convert the point-slope form of a line to the slope-intercept form of a line?

A: Yes, you can convert the point-slope form of a line to the slope-intercept form of a line by substituting the slope and the y-intercept into the slope-intercept form of the equation.

Q: Can I convert the slope-intercept form of a line to the point-slope form of a line?

A: Yes, you can convert the slope-intercept form of a line to the point-slope form of a line by substituting the slope and a point on the line into the point-slope form of the equation.

Q: What are some common mistakes to avoid when working with the point-slope form and the slope-intercept form of a line?

A: Some common mistakes to avoid when working with the point-slope form and the slope-intercept form of a line include:

  • Not substituting the correct values into the equation
  • Not simplifying the equation correctly
  • Not using the correct notation for the slope and the y-intercept

Q: How do I choose between the point-slope form and the slope-intercept form of a line?

A: You should choose the form of the line that is most convenient for the problem you are working on. If you know the slope and a point on the line, the point-slope form may be more convenient. If you know the slope and the y-intercept, the slope-intercept form may be more convenient.

Q: Can I use the point-slope form and the slope-intercept form of a line to solve systems of linear equations?

A: Yes, you can use the point-slope form and the slope-intercept form of a line to solve systems of linear equations. You can substitute the equations into each other and solve for the variables.

Q: Can I use the point-slope form and the slope-intercept form of a line to graph lines?

A: Yes, you can use the point-slope form and the slope-intercept form of a line to graph lines. You can plot the points on the line and draw a line through them.

Q: Can I use the point-slope form and the slope-intercept form of a line to find the equation of a line that passes through two points?

A: Yes, you can use the point-slope form and the slope-intercept form of a line to find the equation of a line that passes through two points. You can substitute the points into the point-slope form of the equation and solve for the variables.

Q: Can I use the point-slope form and the slope-intercept form of a line to find the equation of a line that passes through a point and has a given slope?

A: Yes, you can use the point-slope form and the slope-intercept form of a line to find the equation of a line that passes through a point and has a given slope. You can substitute the point and the slope into the point-slope form of the equation and solve for the variables.