To sketch this line equation, it would be a good idea to write the equation of the line in the slope-intercept form first.
The equation of the line in the slope-intercept form is given by:
[tex]y=mx+b[/tex]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:
[tex]y-3=\frac{9}{5}(x+1)[/tex]We can multiply both sides of the equation by 5:
[tex]\begin{gathered} 5(y-3)=5\cdot\frac{9}{5}(x+1) \\ 5(y-3)=\frac{5}{5}\cdot9(x+1)\Rightarrow\frac{a}{a}=1,\frac{5}{5}=1 \\ 5(y-3)=9(x+1) \end{gathered}[/tex]Now, we have to apply the distributive property to both sides of the equation:
[tex]\begin{gathered} 5(y-3)=9(x+1) \\ 5y-15=9x+9 \end{gathered}[/tex]Add 15 to both sides of the equation, and then divide by 5:
[tex]\begin{gathered} 5y-15+15=9x+9+15 \\ 5y=9x+24 \\ \frac{5y}{5}=\frac{1}{5}(9x+24) \\ y=\frac{9}{5}x+\frac{24}{5} \end{gathered}[/tex]Therefore, the equation is slope-intercept form is:
[tex]y=\frac{9}{5}x+\frac{24}{5}[/tex]Since we have that the original equation of the line was:
[tex]y-3=\frac{9}{5}(x+1)[/tex]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:
[tex]\begin{gathered} y-y_1=m(x-x_1) \\ y-(3)=\frac{9}{5}(x-(-1)) \\ y-3=\frac{9}{5}(x+1) \end{gathered}[/tex]We need another point to graph the line. We can use the y-intercept obtained before:
[tex]\frac{24}{5}=4.8[/tex]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:
[tex]\begin{gathered} y=0\Rightarrow y=\frac{9}{5}x+\frac{24}{5} \\ 0=\frac{9}{5}x+\frac{24}{5} \\ -\frac{24}{5}=\frac{9}{5}x \\ \frac{5}{9}\cdot(-\frac{24}{5})=\frac{5}{9}\cdot(\frac{9}{5})x \\ -\frac{24}{9}=x \\ x=-\frac{8}{3}\approx-2.66666666667 \end{gathered}[/tex]
