Problem 65 - Intermediate Algebra

Question

Use a nonlinear system of equations to solve each problem. Integer Problem. The product of two integers is \(32,\) and their sum is \(12 .\) Find the integers.

Step-by-Step Solution

Answer

The integers are 4 and 8.

Step 1: Define Variables

Let the two integers be \( x \) and \( y \). We want to find the values of \( x \) and \( y \) such that both equations are satisfied.

Step 2: Set Up Equations

Based on the problem, we have two equations: 1. The product of the integers: \( x \times y = 32 \).2. The sum of the integers: \( x + y = 12 \).

Step 3: Express y in Terms of x

Using the second equation, express \( y \) in terms of \( x \):\[ y = 12 - x \].

Step 4: Substitute and Solve Quadratic Equation

Substitute \( y = 12 - x \) into the first equation:\[ x(12 - x) = 32 \]which simplifies to:\[ 12x - x^2 = 32 \]Rearrange to form a standard quadratic equation:\[ x^2 - 12x + 32 = 0 \].

Step 5: Factor the Quadratic Equation

Factor the quadratic equation:\[ x^2 - 12x + 32 = (x - 4)(x - 8) = 0 \].

Step 6: Solve for x

Set each factor equal to zero and solve for \( x \):\[ x - 4 = 0 \] or \[ x - 8 = 0 \]Thus, \( x = 4 \) or \( x = 8 \).

Step 7: Solve for y

Substitute \( x = 4 \) into \( y = 12 - x \):\[ y = 12 - 4 = 8 \]Substitute \( x = 8 \) into \( y = 12 - x \):\[ y = 12 - 8 = 4 \].

Step 8: Verify Solutions

Verify that both pairs satisfy the original conditions:For \( (x, y) = (4, 8) \):- Product: \( 4 \times 8 = 32 \)- Sum: \( 4 + 8 = 12 \)For \( (x, y) = (8, 4) \):- Product: \( 8 \times 4 = 32 \)- Sum: \( 8 + 4 = 12 \).Both pairs satisfy the original conditions.

Key Concepts

Quadratic Equations

Quadratic equations are fundamental in algebra. They represent polynomial equations of degree two. More simply put, they appear in the form

• \[ ax^2 + bx + c = 0 \]

where \( a \), \( b \), and \( c \) are constants with \( a \) not equaling zero. In solving these equations as in our problem, a process called factoring can often be used if the quadratic is factorable.

Here, the quadratic \( x^2 - 12x + 32 = 0 \) is factored into

• \[ (x - 4)(x - 8) = 0 \]

which provides a straightforward method to find the solutions \( x = 4 \) or \( x = 8 \).

Whenever faced with a quadratic equation, check if factoring is possible first as it is often the simplest method of finding solutions.

Integer Solutions

In many algebra problems, especially those involving real-world applications, we seek integer solutions. Integer solutions are those where the answer is a whole number (including negatives and zero).

In our problem, we are tasked with finding two integers whose product equals 32, and whose sum equals 12. We achieve this by solving the quadratic equation

• \[ x^2 - 12x + 32 = 0 \]

Factorization gives us the pairs \((4, 8)\) and \((8, 4)\), meaning both integers satisfy the problem's conditions.

Understanding integer solutions is crucial because real-world problems often demand whole number outputs. It’s important to remember that integers represent concrete counts of items, quantities, and so forth, which solidifies their frequent necessity in solutions.

Problem-Solving in Algebra

Problem-solving in algebra is all about developing systematic methodologies to break down complex questions into manageable parts. As demonstrated in our exercise, a structured approach simplifies solving nonlinear systems of equations:

• Define your variables.
• Set up equations based on the given conditions.
• Express one variable in terms of the other to simplify.
• Substitute and solve resulting equations efficiently.
• Verify your solutions against the original conditions.

By following these steps, students not only find the solution to the current problem but also develop skills applicable to a wide array of algebraic scenarios. Developing a clear understanding of this logical process builds confidence and improves accuracy in solving mathematical problems.