Solving Equation Pairs with Product and Sum

In algebra, when we are given two conditions involving both the sum and the product of two numbers, solving the system of equations becomes a fascinating challenge. This article explores how to solve such a problem step-by-step, ensuring the results are correct and the process is understandable.

The Problem: Two Numbers Add to 13 and Multiply to 42

Let's consider a typical problem: finding two numbers that add up to 13 and have a product of 42. We can represent these numbers as x and y.

Step 1: Setting Up the Equations

We start with the given conditions:

Equation 1: (x y 13) Equation 2: (xy 42)

Step 2: Expressing One Variable in Terms of the Other

From Equation 1, we can express x as:

[x 13 - y]

Substituting this expression into Equation 2:

[(13 - y)y 42]

This simplifies to:

[y^2 - 13y 42 0]

Step 3: Solving the Quadratic Equation

Now, we need to solve the quadratic equation:

[y^2 - 13y 42 0]

We can use the quadratic formula (y frac{-b pm sqrt{b^2 - 4ac}}{2a}) where (a 1), (b -13), and (c 42). However, in this case, we can factorize the equation:

[y^2 - 13y 42 (y - 7)(y - 6) 0]

Solving for y gives us:

[y 7 quad text{or} quad y 6]

Step 4: Finding the Corresponding Values of x

Substituting y values back into the first equation (x y 13) gives:

When y 7, then x 6 When y 6, then x 7

Thus, the pairs of numbers that satisfy both conditions are 6 and 7.

Verification

Let's verify:

[6 7 13]

[6 times 7 42]

Interesting Observations and Patterns

Let's explore a list of numbers that add up to 13 and ensure the product is always ascending by comparing with the expected results:

12, 1: Product 12 (Ascending, not matching expected pattern) 11, 2: Product 22 (Ascending, but not matching expected pattern) 10, 3: Product 30 (Ascending, not matching expected pattern) 9, 4: Product 36 (Ascending, not matching expected pattern) 8, 5: Product 40 (Ascending, closely matching expected pattern) 7, 6: Product 42 (Matching expected pattern exactly, and solving method confirmed correct)

From this list, we see that the pair (7, 6) and (6, 7) is the solution that matches the product exactly, aligning with algebraic computation.

Additional Example: Solving Another Problem

Problem:

Let's verify using another method. We know:

[x y 13]

[x cdot y 42]

Let's solve for y in terms of x from the first equation:

[x 13 - y]

Substituting into the second equation:

[(13 - y)y 42]

This simplifies to:

[y^2 - 13y 42 0]

Factoring the quadratic equation:

[y^2 - 13y 42 (y - 7)(y - 6)]

Solving for y, we get:

[y 7 quad text{or} quad y 6]

Substituting these values back into the equation (x y 13):

When y 7, then x 6 When y 6, then x 7

Thus, the pairs of numbers are (6, 7) and (7, 6).

Conclusion

By systematically solving the equations and verifying the results, we confidently identified the pair (6, 7) as the solution to the problem: two numbers that add up to 13 and have a product of 42. This example highlights the power of algebraic methods in solving such problems.