Problem 33 - Elementary Algebra

Question

In the following exercises, solve each number word problem. One number is 14 less than another. If their sum is increased by seven, the result is \(85 .\) Find the numbers.

Step-by-Step Solution

Answer

The numbers are 46 and 32.

Step 1: Define the variables

Let the first number be represented by the variable, x. Since one number is 14 less than another, the second number can be represented by the expression, x - 14.

Step 2: Set up the equation

According to the problem, the sum of these numbers, when increased by 7, equals 85. So, the equation to represent this relationship is: (x) + (x - 14) + 7 = 85

Step 3: Simplify the equation

Combine like terms on the left-hand side of the equation: x + (x - 14) + 7 = 85 2x - 7 = 85

Step 4: Solve for x

Add 7 to both sides of the equation: 2x - 7 + 7 = 85 + 7 2x = 92 Divide both sides by 2: x = 46

Step 5: Find the second number

Recall that the second number is 14 less than the first number: Second number = 46 - 14 = 32

Step 6: Verify the solution

Check if the sum of the numbers increased by 7 equals 85: 46 + 32 + 7 = 85 85 = 85 The solution is verified.

Key Concepts

solving equations

Solving equations is a fundamental skill in algebra. It involves finding the value of the variable that makes the equation true. Here’s a step-by-step guide for solving equations:

• Start by writing down the given equation.
• Simplify both sides if needed by combining like terms or using distributive properties.
• Isolate the variable on one side using inverse operations (addition, subtraction, multiplication, division).
• Continue simplifying until you solve for the variable.

In our example, the equation was (x) + (x - 14) + 7 = 85.
We simplified this equation step by step to solve for x. This systematic approach is key to solving any algebraic equation.

defining variables

Defining variables helps translate real-world problems into mathematical equations. It’s crucial to clearly define what each variable represents:

• Identify the unknowns in the problem.
• Assign variables to these unknowns.
• Express relationships between the variables using mathematical expressions.

For the given exercise, we defined:

• Let the first number be represented by the variable, x.
• Since one number is 14 less than another, the second number can be represented by the expression, x - 14.

This clear definition allowed us to set up our equation precisely.

simplifying equations

Simplifying equations makes them easier to solve. It involves performing operations to combine like terms and make the equation more manageable:

• Combine like terms (terms with the same variable).
• Perform arithmetic operations to simplify the expression on both sides.

In our exercise, we simplified the equation: (x) + (x - 14) + 7 = 85 to 2x - 7 = 85.
By simplifying, we reduced the complex equation to a basic one, allowing an easier path to solve for x.

verification of solutions

Verification of solutions ensures that the values obtained from solving the equation are correct and satisfy the original problem:

• Plug the solution back into the original equation.
• Check if both sides of the equation are equal.

In our exercise, we verified the solutions by substituting the numbers back into the conditions stated:
46 + 32 + 7 = 85.
Since both sides of the equation matched, we confirmed our solution was accurate. Verification is a crucial step to avoid mistakes and validate your results.