Problem 39 - Elementary Algebra

Question

In the following exercises, solve each number word problem. The sum of two numbers is 14 One number is two less than three times the other. Find the numbers.

Step-by-Step Solution

Answer

The two numbers are 4 and 10.

Step 1: Define Variables

Let's define the two numbers. Let the first number be denoted by \( x \), and the second number be denoted by \( y \).

Step 2: Set Up Equations

The problem states two conditions: 1) The sum of two numbers is 14, and 2) One number is two less than three times the other. This can be written as two equations: \[ x + y = 14 \] \[ y = 3x - 2 \]

Step 3: Substitute

Substitute the expression for \( y \) from the second equation into the first equation: \[ x + (3x - 2) = 14 \]

Step 4: Solve for x

Combine like terms and solve for \( x \): \[ x + 3x - 2 = 14 \] \[ 4x - 2 = 14 \] \[ 4x = 16 \] \[ x = 4 \]

Step 5: Find y

Now that we know \( x = 4 \), substitute \( x \) back into the second equation to find \( y \): \[ y = 3(4) - 2 \] \[ y = 12 - 2 \] \[ y = 10 \]

Step 6: Solution

The two numbers are 4 and 10. We can verify this by checking their sum and the given conditions.

Key Concepts

solving equations

When solving equations, we aim to find the values of unknowns, which are typically represented as variables like \( x \) and \( y \).
In a number word problem, the information given in words is translated into mathematical equations.
This translation helps us use algebraic techniques to find the unknown values.

The steps generally involve:

• Defining variables for the unknown quantities.
• Setting up equations based on the problem's conditions.
• Solving the equations using various methods.
• Substituting back to find all unknown values.
• Verifying the solution to ensure it meets the original problem's conditions.

Understanding each step is crucial as it builds a framework for approaching more complex problems.

linear equations

Linear equations are algebraic expressions where each term is either a constant or the product of a constant and a single variable. They form straight lines when graphed on the coordinate plane.

In this problem, we have two linear equations:

• \( x + y = 14 \)
• \( y = 3x - 2 \)

The powers of the variables are all one, indicating linearity.

These equations work together to describe the relationship between the numbers. By solving them, we can determine the values of \( x \) and \( y \) that satisfy both conditions simultaneously. Linear equations are foundational in algebra and are used extensively in various fields.

substitution method

The substitution method is a technique used to solve systems of equations. It involves solving one equation for one variable and then substituting this expression into another equation.

In our exercise, we first expressed \( y \) in terms of \( x \) using the second equation:

• \( y = 3x - 2 \)

Then, we substituted this expression into the first equation:

• \( x + (3x - 2) = 14 \)

This substitution simplifies the problem to a single equation with one variable, making it easier to solve. By mastering the substitution method, one can efficiently address a variety of algebraic problems.

algebraic expressions

Algebraic expressions are combinations of variables, numbers, and operations. They form the backbone of algebraic equations and are crucial for modeling real-world problems.

In our problem, the algebraic expressions are:

• \( x + y \)
• \(3x - 2 \)
• \(4x - 2 \)

Each of these represents a specific relationship between the variables. Understanding how to manipulate and combine these expressions is essential for solving equations and finding unknowns.

Recognizing patterns and practicing with various problems helps in developing a strong grasp of algebraic expressions, making more complex math more accessible.