Chapter 4.1, Problem 64HP

To determine

To Find:

Summarize the characteristics that the graphs y = 2x + 3, y = 4x + 3, y = -x + 3, and

y = 10x + 3 have in common.

Answer to Problem 64HP

All these equations have y -intercept equal and that is 3.

Explanation of Solution

Formula Used:

Equation of slope-intercept form of a line is given by:

  $y=mx+b\mathrm{............................}(1)$

Where

  $\begin{array}{l}b=y-\mathrm{int}ercept\\ m=\frac{rise}{run}=slope\end{array}$

Graph:

First we plot all the four equations on the graph:

  $\begin{array}{l}y=2x+\mathrm{3......................}(a)\\ y=4x+\mathrm{3......................}(b)\\ y=10x+\mathrm{3.....................}(c)\\ y=-x+\mathrm{3.......................}(d)\end{array}$

(a)

Comparing with equation (1)

We have

  $\begin{array}{l}m=\frac{rise}{run}=\frac{2}{1}\\ b=y-\mathrm{int}ercept=3\end{array}$

So, for plotting the line, first we plot the point (0,3) , then rise 2 units and run 1 unit to plot the point (1,5). Joint the points to get the line on graph.

(b)

Comparing with equation (1)

We have

  $\begin{array}{l}m=\frac{rise}{run}=\frac{4}{1}\\ b=y-\mathrm{int}ercept=3\end{array}$

So, for plotting the line, first we plot the point (0,3) , then rise 4 units and run 1 unit to plot the point (1,7). Joint the points to get the line on graph.

(c)

Comparing with equation (1)

We have

  $\begin{array}{l}m=\frac{rise}{run}=\frac{10}{1}\\ b=y-\mathrm{int}ercept=3\end{array}$

So, for plotting the line, first we plot the point (0,3) , then rise 10 units and run 1 unit to plot the point (1,13). Joint the points to get the line on graph.

(d)

Comparing with equation (1)

We have

  $\begin{array}{l}m=\frac{rise}{run}=-\frac{1}{1}\\ b=y-\mathrm{int}ercept=3\end{array}$

So, for plotting the line, first we plot the point (0,3) , then go down 1 unit and go right 1 unit to plot the point (1,2). Joint the points to get the line on graph.

  

Calculation:

Clearly, from the graph , we can see that all the four lines cut each other at point (0,3).

So,

All these equations have y -intercept equal and that is 3.

They all cross y- axis at 3.