Chapter T, Problem 1BDT

(a)

To determine

To find: The equation of the line that passes through the point $\left(2,-5\right)$.

Answer to Problem 1BDT

The equation of the line is $y=-3x+1$.

Explanation of Solution

Given:

The slope of the line is $-3$.

Formula used:

The equation of the straight line is $y=mx+c$, where m is the slope of the line, c is the x-intercept.

Calculation:

Suppose, the equation of the straight line is $y=mx+c$.

It is given that, the slope of the line is $-3$.

Thus, substitute $m=-3$ in $y=mx+c$ to obtain the equation of the straight line,

$y=-3x+c$

It is given that, the line passes through the point $\left(2,-5\right)$.

Thus, substitute $x=2$ and $y=-5$ in $y=-3x+c$ to obtain the equation of the straight line,

$\begin{array}{l}-5=-3\times 2+c\\ c=-5+6\\ c=1\end{array}$

Thus, substitute $c=1$ in $y=-3x+c$ to obtain the equation of the straight line,

$y=-3x+1$

Thus, the equation of the line is $y=-3x+1$.

(b)

To determine

To find: The equation of the line that passes through the point $\left(2,-5\right)$.

Answer to Problem 1BDT

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

Explanation of Solution

Given:

The line is parallel to the x-axis.

Formula used:

The equation of the straight line that is parallel to the x-axis is $y=a$, where a is a constant.

Calculation:

It is given that the line is parallel to the x-axis.

Suppose, the equation of the straight line is $y=a$.

It is given that, the line passes through the point $\left(2,-5\right)$.

Thus, substitute $x=2$ and $y=-5$ in $y=a$ to obtain the equation of the straight line,

$y=-5$

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

(c)

To determine

To find: The equation of the line that passes through the point $\left(2,-5\right)$.

Answer to Problem 1BDT

The equation of the line is $x=2$.

Explanation of Solution

Given:

The line is parallel to the y-axis.

Formula used:

The equation of the straight line that is parallel to the y-axis is $x=a$, where a is a constant.

Calculation:

It is given that the line is parallel to the y-axis.

Suppose, the equation of the straight line is $x=a$.

It is given that, the line passes through the point $\left(2,-5\right)$.

Thus, substitute $x=2$ and $y=-5$ in $x=a$ to obtain the equation of the straight line,

$x=2$

Thus, the equation of the line is $x=2$.

(d)

To determine

To find: The equation of the line that passes through the point $\left(2,-5\right)$.

Answer to Problem 1BDT

The equation of the line is $y=\frac{1}{2}x-6$.

Explanation of Solution

Given:

The line is parallel to the line $2x-4y=3$.

Formula used:

The equation of the straight line is $y=mx+c$, where m is the slope of the line, c is the x-intercept.

Calculation:

Suppose, the equation of the straight line is $y=mx+c$.

It is given that, the line is parallel to the line $2x-4y=3$, that is $y=\frac{1}{2}x-\frac{3}{4}$.

The slope of the line $2x-4y=3$ is $\frac{1}{2}$.

Thus, substitute $m=\frac{1}{2}$ in $y=mx+c$ to obtain the equation of the straight line,

$y=\frac{1}{2}x+c$

It is given that, the line passes through the point $\left(2,-5\right)$.

Thus, substitute $x=2$ and $y=-5$ in $y=\frac{1}{2}x+c$ to obtain the equation of the straight line,

$\begin{array}{l}-5=\frac{1}{2}\times 2+c\\ c=-5-1\\ c=-6\end{array}$

Thus, substitute $c=-6$ in $y=\frac{1}{2}x+c$ to obtain the equation of the straight line,

$y=\frac{1}{2}x-6$

Thus, the equation of the line is $y=\frac{1}{2}x-6$.