Problem 29 - Elementary Algebra

Question

In the following exercises, solve each number word problem. The sum of two numbers is -45. One number is nine more than the other. Find the numbers.

Step-by-Step Solution

Answer

The numbers are -27 and -18.

Step 1: Define the variables

Let the first number be denoted as let x be the first number. The second number is nine more than the first number, which can be written as x + 9.

Step 2: Set up the equation

According to the problem, the sum of the two numbers is -45. Therefore, we can write the equation as follows: x + (x + 9) = -45.

Step 3: Simplify and solve the equation

Combine like terms and solve for x: 2x + 9 = -45. Subtract 9 from both sides: 2x = -54. Divide both sides by 2: x = -27.

Step 4: Find the second number

Since the second number is nine more than the first number: x + 9 = -27 + 9 = -18. Therefore, the second number is -18.

Step 5: Verify the solution

Add the two numbers to ensure they sum to -45: -27 + (-18) = -45. The solution is correct.

Key Concepts

Solving Equations

Solving equations is a fundamental part of mathematics. It's the process of finding the value of an unknown variable that makes the equation true. Let's break down the steps involved:

• **Step 1:** Write down the equation provided in the word problem. Here, we have the sum of two numbers is -45, so we start with the equation \(x + (x + 9) = -45\).
• **Step 2:** Combine like terms. In our example, we combine the \(x\) terms, giving us \(2x + 9 = -45\).
• **Step 3:** Isolate the variable. Subtract 9 from both sides, leaving \(2x = -54\).
• **Step 4:** Solve for the variable by dividing both sides by 2, resulting in \(x = -27\).

By following these steps, we effectively solve the equation and find the value of the unknown variable.

Defining Variables

Defining variables is a crucial step in solving word problems. It's about translating the written problem into mathematical terms.
In our problem, we're dealing with two numbers. Let's define these numbers:

• **First Number:** Let's call the first number \(x\). This is our initial unknown variable.
• **Second Number:** According to the problem, the second number is nine more than the first number. So, we define the second number as \(x + 9\).

By defining variables, we transform the word problem into an equation we can solve. Clear definitions help clarify the relationships between different elements of the problem.

Simplifying Equations

Simplifying equations is about making them as straightforward as possible so that we can solve them easily.
Once we have our equation \(x + (x + 9) = -45\), the next step is to simplify:

• **Combine Like Terms:** Add the \(x\) terms together. Here, \(x + x = 2x\). So the equation becomes \(2x + 9 = -45\).
• **Isolate the Variable:** Subtract 9 from both sides. This helps to isolate the term with our variable on one side of the equation. So, \(2x = -54\).
• **Solve for the Variable:** Finally, divide both sides by the coefficient of the variable (which is 2 in this case). Thus, \(x = -27\).

Simplification helps us systematically reduce the problem to a solvable form.

Verification of Solutions

Verification of solutions ensures that our answer is correct. This step is often overlooked but is very important. Let's verify our solution:

• **First Number:** From our equation, we found \(x = -27\).
• **Second Number:** We calculate the second number as \(x + 9 = -27 + 9 = -18\).
• **Check the Sum:** Add the two numbers to verify the sum. \(-27 + (-18) = -45\). Since this matches the original condition of the problem, our solution is verified.

Verification ensures that each step was done correctly and the final answer meets the conditions specified in the problem.