Question: Find the $ y $-intercept of the line passing through the points $ (3, 7) $ and $ (9, 19) $.
["How to Find the $ y $-Intercept of a Line Using Two Points: A Step-by-Step Guide", "Finding the $ y $-intercept of a line is a fundamental skill in algebra and coordinate geometry. Whether you're preparing for exams or tackling math problems, understanding how to determine this key feature helps strengthen your curriculum mastery. In this article, we'll explore how to find the $ y $-intercept of the line that passes through the points $ (3, 7) $ and $ (9, 19) $.", "---", "### Step 1: Understand What the $ y $-Intercept Is", "The $ y $-intercept is the point where the line crosses the $ y $-axis. At this point, the $ x $-coordinate is 0, and the corresponding $ y $-coordinate is the $ y $-intercept value (often denoted as $ (0, b) $).", "---", "### Step 2: Calculate the Slope of the Line", "Before finding the $ y $-intercept, we need the equation of the line—starting with its slope. The slope $ m $ is calculated using the formula:", "[\nm = \frac{y_2 - y_1}{x_2 - x_1}\n]", "Substitute the points $ (3, 7) $ and $ (9, 19) $:", "[\nm = \frac{19 - 7}{9 - 3} = \frac{12}{6} = 2\n]", "The slope of the line is $ 2 $, meaning the line rises 2 units vertically for every 1 unit it moves horizontally.", "---", "### Step 3: Use Point-Slope Form to Find the Equation", "Now that we know the slope, we can write the equation of the line using point-slope form:", "[\ny - y_1 = m(x - x_1)\n]", "Using the point $ (3, 7) $:", "[\ny - 7 = 2(x - 3)\n]", "Simplify to slope-intercept form $ y = mx + b $:", "[\ny - 7 = 2x - 6\n]\n[\ny = 2x + 1\n]", "---", "### Step 4: Identify the $ y $-Intercept", "From the equation $ y = 2x + 1 $, we see that the $ y $-intercept occurs when $ x = 0 $:", "[\ny = 2(0) + 1 = 1\n]", "Therefore, the $ y $-intercept is the point $ (0, 1) $.", "---", "### Conclusion: Why Finding the $ y $-Intercept Matters", "Finding the $ y $-intercept helps in graphing lines accurately, interpreting real-world linear relationships, and solving systems of equations. With just two points, you can follow a straightforward process—calculate the slope, write the line equation, and isolate $ y $ when $ x = 0 $.", "Key Takeaway:\nTo find the $ y $-intercept of a line through two points:\n1. Compute the slope.\n2. Use point-slope form to derive $ y = mx + b $.\n3. Set $ x = 0 $ to find $ y $, the $ y $-intercept.", "Now you're ready to confidently solve similar problems and boost your math confidence!", "---", "Keywords:\n$ y $-intercept line through two points, slope-intercept form, find $ y $-intercept, coordinate geometry, algebra problem-solving, line equation derivation.", "Meta Description:\nLearn how to find the $ y $-intercept of a line passing through $ (3, 7) $ and $ (9, 19) $ using slope and equation derivation. A step-by-step guide for students and math enthusiasts."]
