Substitute \( x+y = 13 \): \( (x-y)(13) = 45 \).

["## Solving the Equation Substitute ( x + y = 13 ): ( (x - y)(13) = 45 )", "Understanding how to substitute and solve equations involving variables isn’t just a basic algebra skill—it’s a powerful tool for simplifying real-world problems. One interesting equation that combines substitution with linear relationships is:", "[\n(x - y)(13) = 45 \quad \ ext{where} \quad x + y = 13\n]", "In this article, we’ll explore how to solve this equation step-by-step, understand its substitution method, and see its applications—especially focusing on working with the constraint ( x + y = 13 ).", "---", "### Understanding the Equation Structure", "The equation combines a substitution condition with a multiplicative relationship:", "[\n(x - y)(13) = 45\n]", "Here, ( x + y = 13 ) defines a linear relationship between variables ( x ) and ( y ), while the product ( (x - y) \cdot 13 = 45 ) gives us a specific numerical constraint. Solving these together allows us to find exact values for ( x ) and ( y ), which can be useful in algebra, physics, economics, and engineering contexts.", "---", "### Step-by-Step Solution", "#### Step 1: Solve for ( (x - y) )", "Divide both sides of the equation by 13 to isolate ( x - y ):", "[\nx - y = \frac{45}{13}\n]", "#### Step 2: Write a system of equations", "Now we have a system:", "[\n\begin{cases}\nx + y = 13 \\nx - y = \frac{45}{13}\n\end{cases}\n]", "This system lets us solve for both ( x ) and ( y ) using addition or substitution.", "#### Step 3: Add the two equations", "Adding both equations cancels ( y ):", "[\n(x + y) + (x - y) = 13 + \frac{45}{13}\n]", "[\n2x = 13 + \frac{45}{13}\n]", "Convert 13 to a fraction with denominator 13:", "[\n13 = \frac{169}{13}\n]", "Now add:", "[\n2x = \frac{169}{13} + \frac{45}{13} = \frac{214}{13}\n]", "Divide by 2:", "[\nx = \frac{214}{13 \ imes 2} = \frac{107}{13}\n]", "#### Step 4: Substitute back to solve for ( y )", "Use ( x + y = 13 ):", "[\n\frac{107}{13} + y = 13\n]", "[\ny = 13 - \frac{107}{13}\n]", "Write 13 as ( \frac{169}{13} ):", "[\ny = \frac{169}{13} - \frac{107}{13} = \frac{62}{13}\n]", "---", "### Final Values", "[\nx = \frac{107}{13}, \quad y = \frac{62}{13}\n]", "These fractional solutions confirm that ( x + y = 13 ) and", "[\n(x - y)(13) = \left( \frac{107 - 62}{13} \right)(13) = \frac{45}{13} \cdot 13 = 45\n]", "which satisfies the original equation.", "---", "### Why This Substitution Method Works", "Using substitution with the given equation lets us transform a system of linear constraints into a solvable linear equation. The strategy applies broadly:", "- When dealing with symmetric equations like ( x + y = c ), isolating one variable simplifies substitution.\n- Multiplying or dividing both sides of an equation enables isolating expressions like ( x - y ).\n- Combining such equations allows solving pairs of unknowns, a common skill in optimization, physics, and systems modeling.", "---", "### Practical Applications", "This method isn’t just academic—here are real-world uses:\n- In engineering, solving for two forces or voltages constrained by energy conservation.\n- In economics, analyzing two correlated market variables with fixed total expenditures.\n- In geometry, finding points that maintain a fixed sum of distances while restricting coordinate sums.", "---", "### Conclusion", "The equation ( (x - y)(13) = 45 ), constrained by ( x + y = 13 ), exemplifies how substitution simplifies urgent algebraic problems. By isolating ( x - y ) and solving the linear system, we find exact values for ( x ) and ( y ). Mastering such substitutions strengthens analytical reasoning in both theoretical math and applied sciences.", "Whether you're balancing equations in chemistry, optimizing resource allocation, or modeling physical systems, knowing how to substitute and solve constrained equations is invaluable.", "---", "Keywords: substitute ( x + y = 13 ), solve ( (x - y)(13) = 45 ), algebra substitution method, linear equations with constraints, solving systems of equations, fractional solutions, real-world applications of algebra."]