Solution: The slope $ m $ of the line is $ \frac{19 - 7}{9 - 3} = \frac{12}{6} = 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 $ \boxed{1} $.

February 22, 2026

$Solution: The slope $ m $ of the line is $ \frac{19 - 7}{9 - 3} = \frac{12}{6} = 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 $ \boxed{1} $.$

How to Find the Equation of a Line Using Point-Slope Form: A Step-by-Step Guide

Learning how to determine the equation of a line is a fundamental skill in algebra, essential for graphing and analyzing linear relationships. One powerful method is using the point-slope form, which makes it simple to construct a line when you know a point it passes through and its slope. In this article, we’ll explore how to derive the equation of a line step-by-step, using a real-world example to illustrate the entire process—and how to identify the $ y $-intercept accurately.

Step 1: Understand the Point-Slope Form

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) $

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: $$ y = 2x - 6 + 7 $$ $$ y = 2x + 1 $$

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!