5Question: Find the intersection of the lines $ 4x - 5y = 20 $ and $ 2x + 3y = -6 $.

["5Question: How to Find the Intersection of Two Lines Using System of Equations", "Finding the intersection point of two lines is a fundamental problem in algebra and geometry. Graphically, it represents the point where both lines meet on the coordinate plane. In this article, we’ll explore the step-by-step method to solve the system of equations:", "$$\n\begin{align}\n\ ext{Equation 1:} & \quad 4x - 5y = 20 \\n\ ext{Equation 2:} & \quad 2x + 3y = -6\n\end{align}\n$$", "By solving this system, we’ll discover the exact coordinates of the intersection point.", "---", "### Why Find the Intersection of Lines?", "Solving for the intersection of two lines helps answer real-world problems like determining meeting points, optimizing resources, or finding signals of change in patterns. Whether you're a student learning geometry or a professional in data analysis, mastering this technique is essential.", "---", "### Step 1: Write Down the Equations", "We begin with:\n$$\n(1)\quad 4x - 5y = 20\n$$\n$$\n(2)\quad 2x + 3y = -6\n$$", "---", "### Step 2: Choose a Method — Substitution or Elimination", "For this example, we’ll use the elimination method because it’s efficient with small coefficients.", "---", "### Step 3: Align Coefficients for Elimination", "Our goal is to eliminate one variable (e.g., $ x $) to solve for $ y $ first.", "Multiply Equation (2) by 2 so the $ x $-coefficients match:", "$$\n2 \ imes (2x + 3y) = 2 \ imes (-6) \Rightarrow 4x + 6y = -12\n$$", "Now rewrite the system:\n$$\n\begin{align}\n4x - 5y &= 20 \quad \ ext{(1)} \\n4x + 6y &= -12 \quad \ ext{(2a)}\n\end{align}\n$$", "---", "### Step 4: Subtract Equations to Eliminate $ x $", "Subtract Equation (1) from Equation (2a):", "$$\n(4x + 6y) - (4x - 5y) = -12 - 20\n\Rightarrow 4x + 6y - 4x + 5y = -32\n\Rightarrow 11y = -32\n$$", "Solve for $ y $:", "$$\ny = \frac{-32}{11}\n$$", "---", "### Step 5: Substitute Back to Find $ x $", "Now plug $ y = -\frac{32}{11} $ into Equation (2):\n$$\n2x + 3\left(-\frac{32}{11}\right) = -6\n\Rightarrow 2x - \frac{96}{11} = -6\n$$", "Move constant term to the right:", "$$\n2x = -6 + \frac{96}{11}\n= \frac{-66 + 96}{11} = \frac{30}{11}\n$$", "Solve for $ x $:", "$$\nx = \frac{30}{11} \div 2 = \frac{15}{11}\n$$", "---", "### Step 6: Write the Intersection Point", "The intersection point is the ordered pair:", "$$\n\left( \frac{15}{11}, -\frac{32}{11} \right)\n$$", "---", "### Verification (Important!)", "To confirm, substitute $ x = \frac{15}{11} $, $ y = -\frac{32}{11} $ into both original equations.", "Check Equation 1:\n$$\n4\left(\frac{15}{11}\right) - 5\left(-\frac{32}{11}\right) = \frac{60}{11} + \frac{160}{11} = \frac{220}{11} = 20 \quad \checkmark\n$$", "Check Equation 2:\n$$\n2\left(\frac{15}{11}\right) + 3\left(-\frac{32}{11}\right) = \frac{30}{11} - \frac{96}{11} = \frac{-66}{11} = -6 \quad \checkmark\n$$", "Both equations are satisfied.", "---", "### Why This Works", "The solution satisfies both equations, confirming the point lies on both lines. The elimination method strategically cancels one variable by aligning coefficients, making it a reliable choice for solving linear systems.", "---", "### Conclusion", "Finding the intersection of two lines through systems of equations is a clear and powerful skill. By following the steps of alignment, elimination, and substitution, you can accurately determine the unique point where two linear equations meet. Whether you're solving algebra homework or analyzing data trends, this method empowers you to solve real-world intersection problems with confidence.", "Try it yourself: Pick another pair of lines, follow the steps, and verify your intersection. You’ve unlocked a key algebraic tool!", "---", "Keywords:\nintersection of lines, solve equations graphically, linear system, elimination method, coordinate geometry, algebra tutorial, find intersection point, $ 4x - 5y = 20 $, $ 2x + 3y = -6 $, step-by-step solution, line intersection, algebra 101."]