Chapter 9, Problem 9CA

(a)

To determine

To fill in values of $m\text{ and }b$ so that each given statement is true considering the equation $y=mx+b$

Answer to Problem 9CA

  $\text{When }m=\underset{\_}{\text{ 1 }}\text{ and }b=\underset{\_}{\text{ 5 }},\text{ the graph of the equation passes through point }\left(-1,4\right)$

Explanation of Solution

Given:

  $\text{When }m=\underset{\_}{}\text{ and }b=\underset{\_}{},\text{ the graph of the equation passes through point }\left(-1,4\right)$

Concept Used:

Assume any slope, say $m=1$ take point $\left(-1,4\right)$ find value of b using given equation $y=mx+b$

Calculation:

  $\begin{array}{l}y=mx+b\\ \text{Subsitute slope }m=1\text{ and given point }x=-1\text{ and }y=\text{4}\\ 4=\text{1}\left(-1\right)+b\\ 4=-1+b\\ 4+1=b\\ b=5\end{array}$

Conclusion:

  $\text{When }m=\underset{\_}{\text{ 1 }}\text{ and }b=\underset{\_}{\text{ 5 }},\text{ the graph of the equation passes through point }\left(-1,4\right)$

(b)

To determine

To fill in values of $m\text{ and }b$ so that each given statement is true considering the equation $y=mx+b$

Answer to Problem 9CA

  $\begin{array}{l}\text{When }m=\underset{\_}{\text{ 1 }}\text{ and }b=\underset{\_}{-5},\\ \text{the graph of the equation has a positive slope and passes through point }\left(-2,-5\right)\end{array}$

Explanation of Solution

Given:

  $\begin{array}{l}\text{When }m=\underset{\_}{}\text{ and }b=\underset{\_}{},\\ \text{the graph of the equation has a positive slope and passes through point }\left(-2,-5\right)\end{array}$

Concept Used:

Assume any positive value of slope, say $m=1$ take point $\left(-2,-5\right)$ find value of b using given equation $y=mx+b$

Calculation:

  $\begin{array}{l}y=mx+b\\ \text{Subsitute slope }m=1\text{ and given point }x=-2\text{ and }y=-5\\ -5=\text{1}\left(-2\right)+b\\ -5=-2+b\\ -5+2=b\\ b=-3\end{array}$

Conclusion:

  $\begin{array}{l}\text{When }m=\underset{\_}{\text{ 1 }}\text{ and }b=\underset{\_}{-5},\\ \text{the graph of the equation has a positive slope and passes through point }\left(-2,-5\right)\end{array}$

(c)

To determine

To fill in values of $m\text{ and }b$ so that each given statement is true considering the equation $y=mx+b$

Answer to Problem 9CA

  $\begin{array}{l}\text{When }m=\underset{\_}{\frac{-1}{4}}\text{ and }b=\underset{\_}{\frac{25}{4}},\\ \text{the graph of the equation is perpendicular to the graph of }y=4x-3\text{ and}\\ \text{passes through the point }\left(1,6\right)\end{array}$

Explanation of Solution

Given:

  $\begin{array}{l}\text{When }m=\underset{\_}{}\text{ and }b=\underset{\_}{},\\ \text{the graph of the equation is perpendicular to the graph of }y=4x-3\text{ and}\\ \text{passes through the point }\left(1,6\right)\end{array}$

Concept Used:

The lines are perpendicular if their slopes are negative reciprocal of each other.

Calculation:

Slope of graph that is perpendicular to $y=mx+b$

  $=4$

The lines are perpendicular if their slopes are negative reciprocal of each other.

Therefore,

Slope of equation $y=mx+b$ , $\left(m\right)=\frac{-1}{4}$

Use value of $m$ and point $(1,6)$ to find value of b

  $\begin{array}{l}y=mx+b\\ \text{Subsitute slope }m=\frac{-1}{4}\text{ and given point }x=1\text{ and y}=6\\ 6=\frac{-1}{4}\left(1\right)+b\\ 6=-\frac{1}{4}+b\\ 6+\frac{1}{4}=b\\ b=\frac{24+1}{4}=\frac{25}{4}\end{array}$

Conclusion:

  $\begin{array}{l}\text{When }m=\underset{\_}{\frac{-1}{4}}\text{ and }b=\underset{\_}{\frac{25}{4}},\\ \text{the graph of the equation is perpendicular to the graph of }y=4x-3\text{ and}\\ \text{passes through the point }\left(1,6\right)\end{array}$