Substitute \( x + y = 10 \) and \( xy = 21 \):
["# Solving the System: Substitute ( x + y = 10 ) and ( xy = 21 )", "Understanding systems of equations is essential in algebra, and one classic example involves substituting the sum and product of two variables. Consider the equations:", "[\nx + y = 10\n]\n[\nxy = 21\n]", "These equations represent a foundational problem in quadratic relationships, often encountered in math education and real-world applications. This article explores how to solve such a system using substitution, interpret the results, and understand its significance.", "## Step 1: Express One Variable in Terms of the Other", "From the first equation:", "[\nx + y = 10\n]", "Solve for ( y ):", "[\ny = 10 - x\n]", "This substitution allows replacing ( y ) in the second equation.", "## Step 2: Substitute into the Product Equation", "Substitute ( y = 10 - x ) into ( xy = 21 ):", "[\nx(10 - x) = 21\n]", "Simplify:", "[\n10x - x^2 = 21\n]", "Rearrange into standard quadratic form:", "[\n-x^2 + 10x - 21 = 0\n]", "Multiply through by (-1) to make coefficients positive:", "[\nx^2 - 10x + 21 = 0\n]", "## Step 3: Solve the Quadratic Equation", "Factor the quadratic:", "[\nx^2 - 10x + 21 = (x - 3)(x - 7) = 0\n]", "Set each factor equal to zero:", "[\nx - 3 = 0 \quad \Rightarrow \quad x = 3\n]\n[\nx - 7 = 0 \quad \Rightarrow \quad x = 7\n]", "Now find corresponding ( y ) values using ( y = 10 - x ):", "- If ( x = 3 ), then ( y = 10 - 3 = 7 )\n- If ( x = 7 ), then ( y = 10 - 7 = 3 )", "## Step 4: Interpret the Solution Set", "The system has two solutions:", "[\n(x, y) = (3, 7) \quad \ ext{and} \quad (x, y) = (7, 3)\n]", "These pairs represent the two values that simultaneously satisfy both equations—a perfect application of substitution in solving simultaneous equations.", "## Why This Matters", "This method demonstrates how systems defined by sum and product relate directly to quadratic equations. Recognizing these patterns helps solve problems in algebra, engineering, economics, and even computer science, where quadratic models frequently arise.", "Whether you're a student mastering algebra or a professional applying math concepts, mastering substitution techniques like this unlocks deeper problem-solving skills.", "Keywords for SEO: substitute (x + y = 10), solve (xy = 21), algebraic substitution method, solve system of equations, quadratic from sum and product, algebra problem solving, solve (x + y = 10) and (xy = 21), real numbers system, math substitution technique.", "---", "Optimizing your approach to systems of equations not only boosts accuracy but enhances your mathematical intuition—essential for advanced learning and practical application."]
