The sum of two numbers is 18 and the product of the two numbers is 56. Find the numbers.
Understand the Problem
The question involves solving two problems related to integers. The first one requires finding two numbers given their sum and product, while the second problem relates to setting up an algebraic equation based on the relationships between integers.
Answer
The two numbers are 4 and 14.
Answer for screen readers
The two numbers are 4 and 14.
Steps to Solve
- Define the Variables
Let $x$ and $y$ represent the two numbers. According to the problem, we have:
$$ x + y = 18 $$ $$ x \cdot y = 56 $$
- Express One Variable in Terms of the Other
From the first equation, express $x$ in terms of $y$:
$$ x = 18 - y $$
- Substitute into the Product Equation
Substitute $x$ in the product equation:
$$ (18 - y) \cdot y = 56 $$
- Expand and Rearrange the Equation
Expanding the equation gives:
$$ 18y - y^2 = 56 $$
Rearranging it yields:
$$ y^2 - 18y + 56 = 0 $$
- Factor the Quadratic Equation
Next, we factor the quadratic:
$$(y - 14)(y - 4) = 0$$
This gives two possible solutions for y:
$$ y = 14 \quad \text{or} \quad y = 4 $$
- Find Corresponding x Values
Using the values of $y$, substitute back to find $x$:
If $y = 14$: $$ x = 18 - 14 = 4 $$
If $y = 4$: $$ x = 18 - 4 = 14 $$
Thus, the pairs are $(4, 14)$ and $(14, 4)$.
The two numbers are 4 and 14.
More Information
The numbers add up to 18 and multiply to 56. This is a classic problem dealt with using systems of equations and quadratic factoring.
Tips
- Forgetting to rearrange the equation correctly after substituting values can lead to incorrect solutions.
- Mistaking the signs during factoring can lead to errors in identifying the correct pairs.
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