Solve the system of equations: \( 2x + 3y = 6 \) and \( 4x - y = 5 \).

["# Solve the System of Equations: ( 2x + 3y = 6 ) and ( 4x - y = 5 )", "Solving systems of equations is a fundamental skill in algebra, widely used in science, engineering, economics, and everyday problem-solving. In this article, we’ll walk step-by-step through solving the system:", "[\n\begin{cases}\n2x + 3y = 6 \\n4x - y = 5\n\end{cases}\n]", "This system consists of two linear equations with two variables, ( x ) and ( y ). We’ll explore multiple methods—substitution, elimination, and matrix techniques—to find the solution. Understanding these approaches builds a strong foundation for tackling more complex systems in advanced mathematics.", "---", "## Why Solve Systems of Equations?", "Systems of equations model real-world relationships where multiple conditions must be satisfied simultaneously. For example, this system could represent:", "- The intersection point of two linear lines (graphically),\n- Equilibrium prices in a market (economic models),\n- Resource allocation problems with constraints.", "Mastering these solutions equips you with tools for optimization, predictive modeling, and analytical reasoning.", "---", "## Methods to Solve the System", "### Method 1: Substitution Method", "This method involves solving one equation for one variable and substituting that expression into the other equation.", "Step 1: Solve the second equation for ( y )\nStart with the second equation:\n[\n4x - y = 5\n]\nAdd ( y ) to both sides and subtract 5:\n[\ny = 4x - 5\n]", "Step 2: Substitute into the first equation\nNow plug ( y = 4x - 5 ) into ( 2x + 3y = 6 ):\n[\n2x + 3(4x - 5) = 6\n]\nDistribute:\n[\n2x + 12x - 15 = 6\n]\nCombine like terms:\n[\n14x - 15 = 6\n]\nAdd 15 to both sides:\n[\n14x = 21\n]\nDivide by 14:\n[\nx = \frac{21}{14} = \frac{3}{2}\n]", "Step 3: Find ( y )\nSubstitute ( x = \frac{3}{2} ) into ( y = 4x - 5 ):\n[\ny = 4 \left( \frac{3}{2} \right) - 5 = 6 - 5 = 1\n]", "Solution:\n[\nx = \frac{3}{2}, \quad y = 1\n]", "---", "### Method 2: Elimination Method", "This technique eliminates one variable by aligning coefficients and subtracting equations.", "Step 1: Align equations\n[\n(1)\quad 2x + 3y = 6\n(2)\quad 4x - y = 5\n]", "Multiply equation (2) by 3 to match coefficients of ( y ):\n[\n3(4x - y) = 3(5) \quad \Rightarrow \quad 12x - 3y = 15\n]", "Step 2: Add equations to eliminate ( y )\nAdd to equation (1):\n[\n(2x + 3y) + (12x - 3y) = 6 + 15\n]\n[\n14x = 21 \quad \Rightarrow \quad x = \frac{21}{14} = \frac{3}{2}\n]", "Step 3: Substitute back to find ( y )\nUse ( x = \frac{3}{2} ) in original ( 4x - y = 5 ):\n[\n4 \cdot \frac{3}{2} - y = 5 \Rightarrow 6 - y = 5 \Rightarrow y = 1\n]", "---", "### Method 3: Matrix ( Elimination as Matrix Operations )", "For larger systems, matrices streamline solutions. Here, we apply augmented matrix methods.", "Write the system as a matrix equation:\n[\n\begin{bmatrix}\n2 & 3 & | & 6 \\n4 & -1 & | & 5\n\end{bmatrix}\n]", "Step 1: Eliminate ( x ) beneath the pivot\nMultiply row 1 by 2:\n[\n[4\quad 6\quad | \quad 12]\n]\nSubtract from row 2:\n[\n(4\quad -1\quad | \quad 5) - (4\quad 6\quad | \quad 12) = [0\quad -7\quad | \quad -7]\n]", "Augmented matrix:\n[\n\begin{bmatrix}\n2 & 3 & | & 6 \\n0 & -7 & | & -7\n\end{bmatrix}\n]", "Step 2: Solve for ( y )\n[\n-7y = -7 \quad \Rightarrow \quad y = 1\n]", "Step 3: Back-substitute for ( x )\nFrom row 2: ( 4x - y = 5 ), plug ( y = 1 ):\n[\n4x - 1 = 5 \Rightarrow 4x = 6 \Rightarrow x = \frac{3}{2}\n]", "---", "## Verifying the Solution", "Plug ( x = \frac{3}{2} ), ( y = 1 ) into both original equations:", "First equation:\n[\n2\left( \frac{3}{2} \right) + 3(1) = 3 + 3 = 6 \quad \checkmark\n]", "Second equation:\n[\n4\left( \frac{3}{2} \right) - 1 = 6 - 1 = 5 \quad \checkmark\n]", "The solution satisfies both equations.", "---", "## Graphical Interpretation", "Plotting the two lines:", "- Line 1: ( 2x + 3y = 6 ) → ( y = -\frac{2}{3}x + 2 )\n- Line 2: ( 4x - y = 5 ) → ( y = 4x - 5 )", "They intersect at the point ( \left( \frac{3}{2}, 1 \right) ), confirming the algebraic solution.", "---", "## Key Takeaways", "- The unique solution ( x = \frac{3}{2}, y = 1 ) means the lines intersect at a single point.\n- Systems with no solution (parallel lines) or infinite solutions (coincident lines) indicate"]