Question:** Find the \( y \)-intercept of the line given by the equation \( 3x - 4y = 12 \).
["How to Find the ( y )-Intercept of the Line: ( 3x - 4y = 12 )", "Understanding how to find the ( y )-intercept of a linear equation is a foundational skill in algebra and essential for graphing linear functions. The ( y )-intercept is the point where the line crosses the ( y )-axis — at this point, the value of ( x ) is zero. In this article, we’ll explore step-by-step how to determine the ( y )-intercept of the equation ( 3x - 4y = 12 ).", "---", "### What Is the ( y )-Intercept?", "The ( y )-intercept occurs when ( x = 0 ). To find it, substitute ( x = 0 ) into the equation and solve for ( y ). The resulting ( y )-value is the ( y )-intercept, written as the point ( (0, y) ).", "---", "### Step-by-Step: Finding the ( y )-Intercept of ( 3x - 4y = 12 )", "1. Start with the given equation:\n [\n 3x - 4y = 12\n ]", "2. Substitute ( x = 0 ):\n [\n 3(0) - 4y = 12\n ]\n Simplifies to:\n [\n -4y = 12\n ]", "3. Solve for ( y ):\n Divide both sides by (-4):\n [\n y = \frac{12}{-4} = -3\n ]", "4. Write the ( y )-intercept:\n Since ( x = 0 ) and ( y = -3 ), the ( y )-intercept is the point:\n [\n (0, -3)\n ]", "---", "### Final Answer", "The ( y )-intercept of the line ( 3x - 4y = 12 ) is ( \boxed{(0, -3)} ).", "---", "### Why Is the ( y )-Intercept Important?", "- It helps locate specific points on the coordinate plane.\n- It simplifies graphing: start drawing the line at the ( y )-intercept and use slope to find additional points.\n- It plays a key role in slope-intercept form ( y = mx + b ), where ( b ) is the ( y )-intercept.", "---", "### Practice Problem", "Try finding the ( y )-intercept of the line:\n( 2x + 5y = 10 )", "- Substitute ( x = 0 ):\n ( 2(0) + 5y = 10 \Rightarrow 5y = 10 \Rightarrow y = 2 )\n- The ( y )-intercept is ( (0, 2) )", "---", "Mastering how to find the ( y )-intercept is a powerful tool in algebra — use this method confidently in exams, graphing exercises, and real-world applications involving linear relationships."]
