We are given the graph of a line and we are asked to determine its equation in point-slope form.
The general form in slope point form of a line is:
[tex]y-y_0=m(x-x_0)[/tex]Where:
[tex]\begin{gathered} m=\text{ slope} \\ (x_0,y_0)\text{ is apoint in the line} \end{gathered}[/tex]to determine the slope we will use the following formula:
[tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex]Where:
[tex](x_1,y_1);(x_2,y_2)=\text{ points on the line}[/tex]We will choose two points on the line from the graph:
[tex]\begin{gathered} (x_1,y_1)=(0,1) \\ (x_2,y_2)=(2,2) \end{gathered}[/tex]Now, we plug in the values in the formula for the slope:
[tex]m=\frac{2-1}{2-0}=\frac{1}{2}[/tex]Now, we substitute the value of the slope in the equation of the line:
[tex]y-y_0=\frac{1}{2}(x-x_0)[/tex]Now, we plug in the first point we choose for the line:
[tex]\begin{gathered} y-1=\frac{1}{2}(x-0) \\ \\ y-1=\frac{1}{2}x \end{gathered}[/tex]And thus we have determined the equation of the line in point-slope form.
The slope-intercept form is the following:
[tex]y=mx+b[/tex]To convert this equation to slope-intercept form, we will take the previous equations and we will add 1 to both sides:
[tex]y=\frac{1}{2}x+1[/tex]And thus we have determined the slope-intercept form of the equations of the line.
