Problem 107 - Elementary Algebra

Question

The sum of two numbers is 15. One number is 3 less than the other. Find the numbers.

Step-by-Step Solution

Answer

The numbers are 6 and 9.

Step 1: Define the Variables

Let the two numbers be denoted as x and y. According to the problem, the sum of the two numbers is 15 and one number is 3 less than the other. We can represent this information with the equations: 1) x + y = 15 2) x = y - 3

Step 2: Substitute the Second Equation into the First

Using the second equation x = y - 3, substitute x in the first equation. This will give us: (y - 3) + y = 15

Step 3: Solve the Equation

Combine like terms and solve for y: 2y - 3 = 15 Add 3 to both sides: 2y = 18 Divide by 2: y = 9

Step 4: Find the Other Number

Now that y is found, use the second equation to find x: x = y - 3 x = 9 - 3 x = 6

Step 5: Check the Solution

Verify that the two numbers found satisfy both conditions. x + y = 6 + 9 = 15 and x = y - 3 6 = 9 - 3 Both conditions are satisfied, so the solution is correct.

Key Concepts

Defining Variables

In any algebraic problem, it's essential to start by defining the variables. In our example, the problem states the sum of two numbers is 15, and one number is 3 less than the other. We introduce two variables, say, \( x \) and \( y \), to represent these numbers. Based on the problem, we write down the given information in algebraic form:

1\. The sum of the two numbers is 15, represented as \( x + y = 15 \).
2\. One number is 3 less than the other, represented as \( x = y - 3 \).

Properly defining variables is the first key step to solving any algebraic problem as it transforms the word problem into a form that we can manipulate mathematically.

Substitution Method

With our equations defined, the next strategy uses the substitution method. This involves solving one of the equations for one variable and then substituting this expression into the other equation:

\( x = y - 3 \)
We substitute \( x \) from the second equation into the first:

\[(y - 3) + y = 15 \].
This substitution simplifies the system of equations and allows us to solve for a single variable, making the solution process easier.

Solving Systems of Equations

Next, we solve the system of equations by isolating one variable. From the substitution step, we have the equation \[: (y - 3) + y = 15 \]
Combining like terms, we get:

\[: 2y - 3 = 15 \]
We then add 3 to both sides to simplify:

\( 2y = 18 \)
Lastly, we divide both sides by 2 to solve for \( y \):

\( y = 9 \)
We now have one of the numbers. To find the other number, we use the equation \( x = y - 3 \):

\( x = 9 - 3 \) thus \( x = 6 \). Therefore, the two numbers are 6 and 9.

Checking Solutions

After solving the equations, it's crucial to verify that the solutions satisfy the original problem conditions. We check both our numbers in the context of the original equations:

1. The sum of the numbers: \[: 6 + 9 = 15, \]
which is true.

2. One number being 3 less than the other: \( 6 = 9 - 3 \), which is also true.

Both conditions hold, confirming our solution is correct. Always remember checking your answers in algebra ensures that errors are caught, and the solution makes logical and mathematical sense.