The Line Given By − 5 X + 2 Y = − 4 -5x + 2y = -4 − 5 X + 2 Y = − 4 Is Dilated By A Scale Factor Of K = 5 K = 5 K = 5 , Centered At The Origin. What Is The Equation Of The Image Of The Line After Dilation?

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Introduction

In geometry, dilation is a transformation that changes the size of a figure. It is a type of similarity transformation that preserves the shape of the figure but changes its size. In this article, we will explore the concept of dilation and how it affects the equation of a line. Specifically, we will examine the line given by $-5x + 2y = -4$ and determine the equation of its image after dilation by a scale factor of $k = 5$, centered at the origin.

Understanding Dilation

Dilation is a transformation that changes the size of a figure by a scale factor. The scale factor is a number that determines how much the figure is enlarged or reduced. In this case, the scale factor is $k = 5$, which means that the line will be enlarged by a factor of 5.

The Effect of Dilation on the Line

When a line is dilated by a scale factor of $k$, the coordinates of its points are multiplied by $k$. This means that the $x$-coordinates and $y$-coordinates of the points on the line are both multiplied by $k$.

Equation of the Image of the Line

To find the equation of the image of the line after dilation, we need to multiply the coefficients of $x$ and $y$ in the original equation by the scale factor $k$. The original equation is $-5x + 2y = -4$. Multiplying the coefficients of $x$ and $y$ by $k = 5$, we get:

$$-25x + 10y = -20$$

This is the equation of the image of the line after dilation.

Proof of the Equation

To prove that the equation $-25x + 10y = -20$ is the correct equation of the image of the line, we can substitute the coordinates of a point on the original line into the equation and show that it satisfies the equation.

Let $(x, y)$ be a point on the original line. Then, we know that $-5x + 2y = -4$. Multiplying both sides of this equation by $k = 5$, we get:

$$-25x + 10y = -20$$

This shows that the point $(x, y)$ satisfies the equation $-25x + 10y = -20$, which means that the equation is correct.

Conclusion

In this article, we have explored the concept of dilation and how it affects the equation of a line. We have shown that the line given by $-5x + 2y = -4$ is dilated by a scale factor of $k = 5$, centered at the origin, and determined the equation of its image after dilation. The equation of the image of the line is $-25x + 10y = -20$.

References

• [1] "Dilation" by Math Open Reference. Retrieved from https://www.mathopenref.com/dilation.html
• [2] "Similarity Transformations" by Khan Academy. Retrieved from https://www.khanacademy.org/math/geometry/similarity-transformations

Further Reading

• "Dilation and Similarity" by IXL. Retrieved from https://www.ixl.com/math/dilation-and-similarity
• "Dilation and Similarity" by Mathway. Retrieved from https://www.mathway.com/subjects/dilation-and-similarity

Glossary

• Dilation: A transformation that changes the size of a figure by a scale factor.
• Scale factor: A number that determines how much a figure is enlarged or reduced.
• Similarity transformation: A transformation that preserves the shape of a figure but changes its size.
  The Line Given by $-5x + 2y = -4$ Under Dilation: Q&A =====================================================

Introduction

In our previous article, we explored the concept of dilation and how it affects the equation of a line. We determined the equation of the image of the line given by $-5x + 2y = -4$ after dilation by a scale factor of $k = 5$, centered at the origin. In this article, we will answer some common questions related to dilation and the equation of the image of the line.

Q: What is dilation?

A: Dilation is a transformation that changes the size of a figure by a scale factor. The scale factor is a number that determines how much the figure is enlarged or reduced.

Q: How does dilation affect the equation of a line?

A: When a line is dilated by a scale factor of $k$, the coefficients of $x$ and $y$ in the equation of the line are multiplied by $k$. This means that the equation of the image of the line is obtained by multiplying the coefficients of $x$ and $y$ in the original equation by the scale factor $k$.

Q: What is the equation of the image of the line given by $-5x + 2y = -4$ after dilation by a scale factor of $k = 5$, centered at the origin?

A: The equation of the image of the line is $-25x + 10y = -20$.

Q: How do I determine the equation of the image of a line after dilation?

A: To determine the equation of the image of a line after dilation, you need to multiply the coefficients of $x$ and $y$ in the original equation by the scale factor $k$.

Q: What is the scale factor?

A: The scale factor is a number that determines how much a figure is enlarged or reduced. In this case, the scale factor is $k = 5$.

Q: How does the scale factor affect the equation of the line?

A: The scale factor affects the equation of the line by multiplying the coefficients of $x$ and $y$ in the equation of the line.

Q: Can I dilate a line by a scale factor of $k = 0$?

A: No, you cannot dilate a line by a scale factor of $k = 0$. A scale factor of $0$ would mean that the line is not changed at all, which is not a dilation.

Q: Can I dilate a line by a scale factor of $k = 1$?

A: Yes, you can dilate a line by a scale factor of $k = 1$. A scale factor of $1$ means that the line is not changed at all, which is a dilation.

Q: What is the effect of dilation on the coordinates of a point on the line?

A: The coordinates of a point on the line are multiplied by the scale factor $k$.

Q: How do I find the coordinates of a point on the image of the line after dilation?

A: To find the coordinates of a point on the image of the line after dilation, you need to multiply the coordinates of the point on the original line by the scale factor $k$.

Conclusion

In this article, we have answered some common questions related to dilation and the equation of the image of the line. We have shown that dilation is a transformation that changes the size of a figure by a scale factor, and that the equation of the image of the line is obtained by multiplying the coefficients of $x$ and $y$ in the original equation by the scale factor $k$. We have also provided examples of how to determine the equation of the image of a line after dilation.

References

• [1] "Dilation" by Math Open Reference. Retrieved from https://www.mathopenref.com/dilation.html
• [2] "Similarity Transformations" by Khan Academy. Retrieved from https://www.khanacademy.org/math/geometry/similarity-transformations

Further Reading

• "Dilation and Similarity" by IXL. Retrieved from https://www.ixl.com/math/dilation-and-similarity
• "Dilation and Similarity" by Mathway. Retrieved from https://www.mathway.com/subjects/dilation-and-similarity

Glossary

• Dilation: A transformation that changes the size of a figure by a scale factor.
• Scale factor: A number that determines how much a figure is enlarged or reduced.
• Similarity transformation: A transformation that preserves the shape of a figure but changes its size.