Problem 38 - Elementary Algebra

Question

In the following exercises, solve each number word problem. One number is six more than five times another. Their sum is six. Find the numbers.

Step-by-Step Solution

Answer

The numbers are 0 and 6.

Step 1: Define Variables

Let one number be denoted as x and another number be denoted as y.

Step 2: Formulate Equations

Translate the word problem into two mathematical equations:1. One number is six more than five times another: y = 5x + 62. Their sum is six: x + y = 6

Step 3: Substitute and Solve for x

Substitute the expression for y from the first equation into the second equation:x + (5x + 6) = 6Combine like terms: 6x + 6 = 6Subtract 6 from both sides:6x = 0Divide by 6:x = 0

Step 4: Solve for y

Now substitute x = 0 back into the first equation to find y:y = 5(0) + 6y = 6

Step 5: Verify the Solution

Check the values by plugging them back into the original word problem:Is y = 6 more than five times x = 0? Yes, because 6 = 6.Do the numbers add up to 6? Yes, 0 + 6 = 6.

Key Concepts

Algebraic Equations

Algebraic equations are mathematical statements that show the equality of two expressions. In this number word problem, we are given conditions about two numbers and we convert these conditions into algebraic equations.

An equation involves variables (unknown values represented by letters) and constants (known values represented by numbers). For instance, in the problem, we use the variables **x** and **y** to represent the two unknown numbers.

To formulate equations:

• Identify the relationships between the numbers as described in the problem.
• Translate these relationships into equalities using algebraic expressions.

In this example, the two equations derived are:

1. **y** = 5**x** + 6
2. **x** + **y** = 6.

These equations are the foundation for solving the problem.

Solving Systems of Equations

Systems of equations involve finding the values of variables that satisfy multiple equations simultaneously. This number word problem involves a system of two linear equations. To solve this system, we employ methods like substitution or elimination.

In this exercise, we use the substitution method. Here's how it works:

• First, solve one of the equations for one variable in terms of the other variable.
• Then, substitute this expression into the other equation.
• Finally, solve for the remaining variable.

By substituting **y** from the first equation into the second equation, we simplify the problem into a single-variable equation:

**x** + (5**x** + 6) = 6 → 6**x** + 6 = 6 → 6**x** = 0 → **x** = 0

Once we find **x**, we substitute it back into the first equation to find **y**:

**y** = 5(0) + 6 → **y** = 6

This method ensures that the solution satisfies both equations.

Variable Substitution

Variable substitution is a key technique in solving systems of linear equations. It involves replacing one variable with an expression that represents it in terms of another variable. This helps to reduce the number of variables in the equation, making it easier to solve.

Here's a step-by-step guide to substitution:

• Isolate one variable in one of the equations.
• Substitute this expression into the other equation.
• Solve the resulting single-variable equation.

Applying this technique to our problem:

1. From **y** = 5**x** + 6 (first equation), we isolate **y**.
2. Substitute **y** in the second equation: **x** + (5**x** + 6) = 6.

This process simplifies it to a single equation in terms of **x**:

6**x** + 6 = 6 → 6**x** = 0 → **x** = 0.

After finding **x**, substitute it back to find **y**:

**y** = 5(0) + 6 → **y** = 6.

This technique helps simplify the process, making it easier to find the solution to the system of equations.