Finding the Equation of the Image of the Line

Oct 13, 2025 by ludoplayers7.com 130 views

Introduction

In mathematics, dilation is a transformation that changes the size of a figure. It is a type of transformation that can be used to enlarge or shrink a figure. In this article, we will discuss how to dilate a line given by the equation $$y=\frac{2}{5} x+3$$ by a scale factor of $$k=\frac{5}{3}$$, centered at the origin.

Understanding Dilation

Dilation is a transformation that changes the size of a figure. It is a type of transformation that can be used to enlarge or shrink a figure. When a figure is dilated, its size is changed, but its shape remains the same. The scale factor of a dilation is a number that tells us how much the figure is enlarged or shrunk.

Dilating a Line

To dilate a line, we need to multiply the coordinates of the line by the scale factor. The scale factor is a number that tells us how much the line is enlarged or shrunk. In this case, the scale factor is $$k=\frac{5}{3}$$.

Finding the Equation of the Image of the Line

To find the equation of the image of the line after dilation, we need to multiply the coordinates of the line by the scale factor. The coordinates of the line are given by the equation $$y=\frac{2}{5} x+3$$. We can rewrite this equation as $$y=\frac{2}{5} x+3$$.

Multiplying the Coordinates by the Scale Factor

To dilate the line, we need to multiply the coordinates of the line by the scale factor. The scale factor is $$k=\frac{5}{3}$$.

\begin{aligned} x'&=kx\\ &=\frac{5}{3}x \end{aligned}

\begin{aligned} y'&=ky\\ &=\frac{5}{3}y \end{aligned}

Substituting the Values of $$x'$$ and $$y'$$ into the Equation of the Line

We can substitute the values of $$x'$$ and $$y'$$ into the equation of the line to get the equation of the image of the line.

\begin{aligned} y'&=\frac{2}{5}x'+3\\ &=\frac{2}{5}\left(\frac{5}{3}x\right)+3\\ &=\frac{2}{3}x+3 \end{aligned}

Conclusion

In this article, we have discussed how to dilate a line given by the equation $$y=\frac{2}{5} x+3$$ by a scale factor of $$k=\frac{5}{3}$$, centered at the origin. We have found the equation of the image of the line after dilation by multiplying the coordinates of the line by the scale factor and substituting the values of $$x'$$ and $$y'$$ into the equation of the line.

Final Answer

The equation of the image of the line after dilation is $$y'=\frac{2}{3}x+3$$.

References

[1] "Dilation" by Math Open Reference. Math Open Reference, 2022.
[2] "Transformation of the Coordinate Plane" by CK-12. CK-12, 2022.

Keywords

Dilation
Scale factor
Coordinate plane
Line
Equation
Transformation
Geometry
Mathematics

Introduction

In our previous article, we discussed how to dilate a line given by the equation $$y=\frac{2}{5} x+3$$ by a scale factor of $$k=\frac{5}{3}$$, centered at the origin. In this article, we will answer some frequently asked questions about dilating a line.

Q: What is dilation?

A: Dilation is a transformation that changes the size of a figure. It is a type of transformation that can be used to enlarge or shrink a figure.

Q: What is a scale factor?

A: A scale factor is a number that tells us how much a figure is enlarged or shrunk.

Q: How do I dilate a line?

A: To dilate a line, you need to multiply the coordinates of the line by the scale factor.

Q: What is the equation of the image of the line after dilation?

A: The equation of the image of the line after dilation is $$y'=\frac{2}{3}x+3$$.

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

A: The coordinates of the line are multiplied by the scale factor.

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

A: Yes, you can dilate a line by a scale factor of 1. This means that the line will not change size.

Q: Can I dilate a line by a negative scale factor?

A: Yes, you can dilate a line by a negative scale factor. This means that the line will be reflected across the origin.

Q: What is the difference between dilation and reflection?

A: Dilation changes the size of a figure, while reflection changes the orientation of a figure.

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

A: No, you cannot dilate a line by a scale factor of 0. This would mean that the line would disappear.

Q: Can I dilate a line by a scale factor of infinity?

A: No, you cannot dilate a line by a scale factor of infinity. This would mean that the line would become infinitely large.

Q: What is the effect of dilation on the slope of the line?

A: The slope of the line is multiplied by the scale factor.

Q: Can I dilate a line by a scale factor that is not a rational number?

A: Yes, you can dilate a line by a scale factor that is not a rational number. This means that the line will be dilated by a non-rational scale factor.

Q: Can I dilate a line by a scale factor that is negative and not a rational number?

A: Yes, you can dilate a line by a scale factor that is negative and not a rational number. This means that the line will be dilated by a negative non-rational scale factor.