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
This question is based on the concept of line equation, where the equation of a line is given as:
\[ y = mx + c \]
Where the slope is represented by $m$ while the intercept of the line is represented by $c$. The slope is a numeric value that shows the line’s inclination and is equivalent to the $\tan$ of the line’s angle with the positive $x-axis$.
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
Images/Mathematical drawings are created with Geogebra.