Chapter 4.1, Problem 35E

To determine

To write equationof the line that represents line $k$ where line $l$ is the reflection of line $k$ in $x$ -axis −

  

Answer to Problem 35E

The equation of the line $k$ is $3y=5x-3$ .

Explanation of Solution

Given:

  Point $\left(x_1,y_1\right)$ of line $l$$=$$\left(0,1\right)$

  Point $\left(x_2,y_2\right)$ of line $l$$=$$\left(3,-4\right)$

Formula Used:

When a point is reflected across the x-axis, the x-coordinate remains the same, but the y-coordinate is transformed into its opposite (its sign is changed).

The equation of a line is written as $y=mx+b$ where m is the slope and $b$ is the $y$ -intercept.

  $m=\frac{y_2-y_1}{x_2-x_1}$

Calculation:

Given:

  Point $\left(x_1,y_1\right)$ of line $l$$=$$\left(0,1\right)$

  Point $\left(x_2,y_2\right)$ of line $l$$=$$\left(3,-4\right)$

Since, line $l$ is the reflection of line $k$ , thus the co-ordinates of line $k$ are −

Point $\left(x_1,y_1\right)$ of line $k$$=$$\left(0,-1\right)$

  Point $\left(x_2,y_2\right)$ of line $k$$=$$\left(3,4\right)$

Thus, slope $=$$m$$=$$\frac{y_2-y_1}{x_2-x_1}$

  $m$$=$$\frac{4-\left(-1\right)}{3-0}$

  $m$$=$$\frac{5}{3}$

As per the above definition, we have equation of the line as

  $y_1=m{x}_1+b$

  $\begin{array}{l}-1=\left(\frac{5}{3}\times 0\right)+b\\ b=-1\end{array}$

Thus, the equation of the line $k$ −

  $\begin{array}{l}y=\left(\frac{5}{3}\times x\right)+\left(-1\right)\\ 3y=5x-3\end{array}$

Thus, the equation of the line $k$ is $3y=5x-3$ .

Conclusion:

Hence, the equation of the line $k$ is $3y=5x-3$ .