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.**

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:

\[ c = 240\ -\ (1.5) (50) \]

\[ c = 240\ -\ 75 \]

\[ c = 165 \]

*Images/Mathematical drawings are created with Geogebra.*