Question: The lines $ 3x + 2y = 6 $ and $ x - y = 1 $ intersect at point $ (a, b) $. Find $ a $.
["Title: Solving the Intersection of Two Lines: Finding the x-Coordinate (a) of Point (a, b)", "Meta Description:\nDiscover how to find the exact value of $ a $, the x-coordinate, where the lines $ 3x + 2y = 6 $ and $ x - y = 1 $ intersect. Follow step-by-step algebra to determine point $ (a, b) $.", "---", "## Finding $ a $: The X-Coordinate Where Two Lines Intersect", "When two linear equations intersect, their point of intersection satisfies both equations simultaneously. This is a fundamental concept in coordinate geometry and algebra, useful for solving systems of equations.", "In this article, we’ll determine the x-coordinate $ a $, where the lines defined by:", "$$\n3x + 2y = 6 \quad \ ext{(Equation 1)}\n$$\n$$\nx - y = 1 \quad \ ext{(Equation 2)}\n$$", "intersect. Once $ a $ is found, we’ll use it to compute the corresponding $ b $, providing the full point $ (a, b) $ of intersection.", "---", "### Step 1: Solve the System Using Substitution", "We begin by solving Equation 2 for $ y $ in terms of $ x $, since this expression is simpler and allows substitution into Equation 1.", "From Equation 2:\n$$\nx - y = 1 \Rightarrow y = x - 1\n$$", "Now substitute $ y = x - 1 $ into Equation 1:", "$$\n3x + 2(x - 1) = 6\n$$", "---", "### Step 2: Simplify and Solve for $ x $", "Expand and simplify:", "$$\n3x + 2x - 2 = 6\n$$\n$$\n5x - 2 = 6\n$$\n$$\n5x = 6 + 2\n$$\n$$\n5x = 8\n$$\n$$\nx = \frac{8}{5}\n$$", "So, the $ x $-coordinate at the intersection is $ a = \frac{8}{5} $.", "---", "### Step 3: Find $ b $ (Optional)", "To complete the intersection point, substitute $ a = \frac{8}{5} $ into $ y = x - 1 $:", "$$\nb = \frac{8}{5} - 1 = \frac{8}{5} - \frac{5}{5} = \frac{3}{5}\n$$", "Thus, the intersection point is $ \left( \frac{8}{5}, \frac{3}{5} \right) $.", "---", "### Why This Matters", "Finding the intersection $ (a, b) $ helps in graphing linear equations, modeling real-world scenarios like budgeting or physics problems, and is foundational for more advanced math topics such as linear programming and calculus.", "---", "Summary:\nThe lines $ 3x + 2y = 6 $ and $ x - y = 1 $ intersect at the point with $ x $-coordinate\n$$\n\boxed{a = \frac{8}{5}}\n$$", "---", "### Frequently Asked Questions (FAQ)", "Q: Why do we substitute $ y = x - 1 $ into the first equation?\nA: Since $ y = x - 1 $ expresses $ y $ directly in terms of $ x $, it simplifies the system by reducing one variable, making it easier to solve for $ x $.", "Q: Can I use elimination instead of substitution?\nA: Yes! You can eliminate $ y $ by manipulating the equations. Multiply Equation 2 by 2:\n$ 2(x - y) = 2(1) \Rightarrow 2x - 2y = 2 $.\nThen add this to $ 3x + 2y = 6 $:\n$ (3x + 2y) + (2x - 2y) = 6 + 2 \Rightarrow 5x = 8 \Rightarrow x = \frac{8}{5} $.\nBoth methods work—choose the one you find easier!", "Q: What does the intersection point tell us geometrically?\nA: It is the unique point where both lines cross on the coordinate plane, representing a simultaneous solution to both equations.", "---", "Keywords: intersecting lines, solve linear equations, find a-coordinate, intersection point (a, b), algebra tutorial, geometry problem, system of equations, eliminate fractions, substitution method, solve for x.\nDuration: ~6–8 minutes reading\nTarget Audience: High school math students, self-learners, educators\nSEO Targets: $ intersection of $ 3x + 2y = 6 \ ext{ and } x - y = 1 $, $ find x in $ 3x + 2y = 6 $ and $ x - y = 1 $, solve linear system, find $ a $ from intersecting lines."]
