a = 1
b = 6
y = 1/3 x + 1
Explanation:To find a and b, we need to find the slope of the equation using any two points on the line
Slope formula:
[tex]m\text{ = }\frac{y_2-y_1}{x_2-x_1}[/tex]Using points: (3, 2) and (12, 5):
[tex]\begin{gathered} x_1=3,y_1=2,x_2=12,y_2\text{ = }5 \\ m\text{ = }\frac{5-2}{12-3}=\frac{3}{9} \\ m\text{ =slope = 1/3} \end{gathered}[/tex]To get a, we would use points (0, a) and (3, 2) in the slope formula:
[tex]\begin{gathered} m\text{ =}\frac{2-a}{3-0} \\ \frac{1}{3}=\frac{2-a}{3} \\ 3(1)\text{ = 3(2-a)} \\ 3\text{ = 6 - 3a} \\ 3\text{ - 6 = -3a} \\ -3\text{ = -3a} \\ \frac{-3}{-3\text{ }}=\frac{-3a}{-3} \\ a\text{ = 1} \end{gathered}[/tex]To get b, we would use points (3, 2) and (b, 3) in the slope formula:
[tex]\begin{gathered} m\text{ = }\frac{3-2}{b-3} \\ \frac{1}{3}\text{= }\frac{1}{b-3} \\ 3(1)\text{ = 1(b-3)} \\ 3\text{ = b - 3} \\ 3+3\text{ = b} \\ b\text{ = 6} \end{gathered}[/tex]The equation of line: y = mx + c
m = 1/3
c = y-intercept
The y-intercept is the value of y when the line crosses the y-axis:
The value of y = a = 1
c = y-intercept = 1
Inserting the values into the equation:
y = 1/3 x + 1
