Problem 183 - Elementary Algebra

Question

In the following exercises, translate to a system of equations and solve the system. The sum of two numbers is fifteen. One number is three less than the other. Find the numbers.

Step-by-Step Solution

Answer

The numbers are 9 and 6.

Step 1 - Define the variables

Let the two numbers be represented by the variables \(x\) and \(y\). Define \(x\) as one of the numbers and \(y\) as the other number.

Step 2 - Set up the equations based on the problem statement

Translate the statements into equations. The first sentence says, 'The sum of two numbers is fifteen.' This can be written as: \[x + y = 15\]. The second sentence says, 'One number is three less than the other.' This can be written as: \[y = x - 3\].

Step 3 - Substitute the second equation into the first

Substitute the expression for \(y\) from the second equation into the first equation. Replace \(y\) in \(x + y = 15\) with \(x - 3\). The equation becomes: \[x + (x - 3) = 15\].

Step 4 - Simplify and solve for \(x\)

Combine like terms and solve for \(x\): \[2x - 3 = 15\]. Add 3 to both sides to get: \[2x = 18\]. Divide both sides by 2 to find: \[x = 9\].

Step 5 - Solve for \(y\)

Use the value of \(x\) to find \(y\). Substitute \(x = 9\) back into the equation \(y = x - 3\), giving: \[y = 9 - 3\], which simplifies to: \[y = 6\].

Step 6 - State the solution

The two numbers that satisfy the given conditions are \(x = 9\) and \(y = 6\). Thus, the numbers are 9 and 6.

Key Concepts

Variable Definition

In any system of equations problem, the first step is defining the variables. This is crucial because variables represent the unknown quantities we are trying to find. In this case, we are dealing with two numbers.
To start, let's define these numbers as variables. Let's call them x and y. For simplicity:
- Let x be one of the numbers.
- Let y be the other number.
By defining the variables clearly, it helps us translate the word problem into mathematical equations.

Equation Setup

Once the variables are defined, the next step is setting up the equations based on the problem statement. This involves translating the given information into algebraic expressions:
- The problem states: 'The sum of two numbers is fifteen.' This gives us the equation \(x + y = 15\).
- It also states: 'One number is three less than the other.' This gives us the equation \(y = x - 3\).
By setting up these equations, we create a system of equations which we can solve to find the values of x and y.

Substitution Method

To solve the system of equations, we can use the substitution method. This involves solving one equation for one variable and then substituting that expression into the other equation:
- From the second equation, we already have y in terms of x: y = x - 3.
- Substitute this expression into the first equation: x + (x - 3) = 15.
Substituting helps to reduce the number of variables in one of the equations, making it easier to solve.

Combining Like Terms

After substituting, we need to simplify the equation by combining like terms:
- The substituted equation is: \(x + (x - 3) = 15\).
- Combine the x terms: \(2x - 3 = 15\).
Now, solve for x:
- Add 3 to both sides: \(2x = 18\).
- Divide both sides by 2: \(x = 9\).
This step is important to isolate the variable and find its value.

Solution Verification

Finally, it's important to verify the solution to ensure it satisfies the original equations:
- We have found that x = 9.
- Substitute x back into the second equation to find y: \(y = x - 3 \Rightarrow y = 9 - 3 \Rightarrow y = 6\).
Check both values in the first equation: x + y = 15:
- \9 + 6 = 15\ which is correct.
Verifying the solution confirms that our found values solve the system of equations correctly. Hence, the two numbers are 9 and 6.