This question aims to find the $y$-intercept from the equation of the line by first finding the slope coefficient. The point at which the graph line crosses the $y-axis$ is known as the $y$-intercept. Figure 1 illustrates the graphical concept of the $y$-intercept.
Figure 1
Expert Answer
The equation of the line is given as: \[ \overline{y} = b_1 \overline{x} + b_0 \] From the given values, we know that: \[ \overline{x} = 57, \hspace{0.4in} \overline{y} = 251, \hspace{0.4in} s_x = 12, \hspace{0.4in} s_y = 37, \hspace{0.4in} r = 0.341 \] To find the $y$-intercept, first, we have to find the slope coefficient. For slope coefficient, the formula is given as: \[ b_1 = r (\dfrac{s_y} {s_x}) \] By putting in the values, we get: \[ b_1 = (0.341) (\dfrac{37} {12}) \] \[ b_1 = (0.341) (3.083) \] \[ b_1 = 1.051 \] Now, the $y$-intercept coefficient is given as: \[ b_o = \overline{y}\ -\ b_1 \overline{x} \] By putting in the values, we get: \[ b_o = 251\ -\ (1.051) (57) \] \[ b_0 = 251\ -\ 59.9 \] \[ b_0 = 191.9 \]Numeric Result
The $y$-intercept of the line with a slope coefficient of $1.051$, $\overline{x} = 57$, and $\overline{y} = 251$ is $191.9$.Example
Find the $y$-intercept if $\overline{x} =50$, $\overline{y} =240$, $s_x=6$, $s_y=30$ and $r=0.3$. The equation of lines is given as: \[ y = mx + c \] From the given values, we know that: \[ \overline{x} = 50, \hspace{0.4in} \overline{y} = 240, \hspace{0.4in} s_x = 6, \hspace{0.4in} s_y = 30, \hspace{0.4in} r = 0.3 \] To find the $y$-intercept, we have to find the slope coefficient. For slope coefficient, we have the formula given as: \[ m = r (\dfrac{s_y} {s_x}) \] By putting in the values, we get: \[ m = (0.3) (\dfrac{30}{6}) \] \[ m = (0.3) (5) \] \[ m = 1.5 \] Now, the $y$-intercept coefficient is: \[ c = y\ -\ mx \] By putting in the values, we get: \[ c = 240\ -\ (1.5) (50) \] \[ c = 240\ -\ 75 \] \[ c = 165 \]
Figure 2