Question

Sketch the graph of the line whose equation , in point-slope form , is y-3 =9/5 (×+1 ). Also write the equation of this line in slope-intercept form . (y=mx+b)

Answer 1

To sketch this line equation, it would be a good idea to write the equation of the line in the slope-intercept form first.

Finding the equation of the line in slope-intercept form

The equation of the line in the slope-intercept form is given by:

`y=mx+b`

Where:

• m is the slope of the line.

,

• b is the y-intercept of the line (the point where the line passes through the y-axis. At this point, x = 0.

Now, we have that the line equation is given in point-slope form as follows:

`y-3=(9)/(5)(x+1)`

We can multiply both sides of the equation by 5:

`\begin{gathered} 5(y-3)=5\cdot(9)/(5)(x+1) \\ 5(y-3)=(5)/(5)\cdot9(x+1)\Rightarrow(a)/(a)=1,(5)/(5)=1 \\ 5(y-3)=9(x+1) \end{gathered}`

Now, we have to apply the distributive property to both sides of the equation:

`\begin{gathered} 5(y-3)=9(x+1) \\ 5y-15=9x+9 \end{gathered}`

Add 15 to both sides of the equation, and then divide by 5:

`\begin{gathered} 5y-15+15=9x+9+15 \\ 5y=9x+24 \\ (5y)/(5)=(1)/(5)(9x+24) \\ y=(9)/(5)x+(24)/(5) \end{gathered}`

Therefore, the equation is slope-intercept form is:

`y=(9)/(5)x+(24)/(5)`

Sketching the graph for the line

Since we have that the original equation of the line was:

`y-3=(9)/(5)(x+1)`

We already know that one of the points of the line is (-1, 3) since the point-slope form of the line is given by:

`\begin{gathered} y-y_1=m(x-x_1) \\ y-(3)=(9)/(5)(x-(-1)) \\ y-3=(9)/(5)(x+1) \end{gathered}`

We need another point to graph the line. We can use the y-intercept obtained before:

`(24)/(5)=4.8`

And since we know it is the y-intercept, we have that this point is (0, 4.8). Therefore, we can graph this equation using the following points:

(0, 4.8) and (-1, 3). Then we can sketch the line as follows:

To have a more precise graph for the line, we can use a graphing calculator:

We can see that the line passes through the x-axis at the point:

`\begin{gathered} y=0\Rightarrow y=(9)/(5)x+(24)/(5) \\ 0=(9)/(5)x+(24)/(5) \\ -(24)/(5)=(9)/(5)x \\ (5)/(9)\cdot(-(24)/(5))=(5)/(9)\cdot((9)/(5))x \\ -(24)/(9)=x \\ x=-(8)/(3)\approx-2.66666666667 \end{gathered}`