Solution: The slope $ m $ of the line is $ \frac19 - 79 - 3 = \frac126 = 2 $. Using point-slope form with point $ (3, 7) $: $ y - 7 = 2(x - 3) $. Simplify: $ y = 2x - 6 + 7 = 2x + 1 $. The $ y $-intercept occurs when $ x = 0 $, so $ y = 2(0) + 1 = 1 $. Thus, the $ y $-intercept is $ \boxed1 $. - Decision Point

Understanding the Context

The point-slope form of a line’s equation is written as:
$$
y - y_1 = m(x - x_1)
$$
where:

• $ m $ is the slope of the line
  
• $ (x_1, y_1) $ is any known point on the line
  
• $ (x, y) $ are the coordinates of any other point on the line

This form is especially useful when you’re given slope and one point, but not in slope-intercept ($ y = mx + b $) or standard form.

Step 2: Identify the Given Values

In this example, we’re told:

• The slope $ m = rac{19 - 7}{9 - 3} = rac{12}{6} = 2 $
  
• A point on the line is $ (3, 7) $

Key Insights

Our goal is to write the equation in slope-intercept form and find the $ y $-intercept.

Step 3: Write the Equation in Point-Slope Form

Substituting $ m = 2 $ and $ (x_1, y_1) = (3, 7) $ into the slope-point formula:
$$
y - 7 = 2(x - 3)
$$

Step 4: Simplify to Slope-Intercept Form

Now simplify the equation to identify $ b $, the $ y $-intercept:
$$
y - 7 = 2x - 6
$$
Add 7 to both sides:
$$
y = 2x - 6 + 7
$$
$$
y = 2x + 1
$$

Final Thoughts

Step 5: Find the $ y $-Intercept

The $ y $-intercept is the value of $ y $ when $ x = 0 $. From the equation $ y = 2x + 1 $, substitute $ x = 0 $:
$$
y = 2(0) + 1 = 1
$$
Thus, the $ y $-intercept is $ oxed{1} $.

Final Thoughts

Mastering point-slope form helps you efficiently model linear equations from minimal data. Once you have the equation, finding the $ y $-intercept is straightforward—just substitute $ x = 0 $. This knowledge supports deeper work in graphing, solving systems of equations, and real-world modeling across science, economics, and engineering.

Practice Tip: Try plugging in $ x = 0 $ into your final equation each time—this quick check confirms your $ y $-intercept is correct!