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Chapter 4: Problem 77

The sum of two numbers is 37 . Their difference is 9. Find the numbers.

Short Answer

Expert verified
The numbers are 23 and 14.

Step by step solution

01

- Set Up Equations

Let the two numbers be represented by variables: let the first number be denoted as x and the second number be denoted as y. According to the problem, the sum of the numbers is 37, so we can write the first equation as: \[ x + y = 37 \] The difference of the numbers is 9, which gives us the second equation: \[ x - y = 9 \]
02

- Solve the System of Equations

To find the values of x and y, we can add the two equations together. Adding equation (1) and equation (2) eliminates y: \[ (x + y) + (x - y) = 37 + 9 \] Simplifying this, we get: \[ 2x = 46 \] Now, solve for x: \[ x = 23 \]
03

- Find the Value of y

Substitute the value of x back into the first equation to find y: \[ 23 + y = 37 \] Solving for y, we get: \[ y = 14 \]
04

- Verify the Solution

To check if the values are correct, substitute x and y back into the original equations: Sum: \[ 23 + 14 = 37 \] Difference: \[ 23 - 14 = 9 \] Both conditions are satisfied, so the solution is correct.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Sum of Numbers
When dealing with problems involving the sum of numbers, it's important to set up an equation that represents this relationship. In our given exercise, we are told that the sum of two numbers is 37.
This can be translated into the equation:

  • x + y = 37
where x and y are the unknown numbers we need to find. Breaking this down:
  • x represents one number.
  • y represents the other number.
Forming an equation from the sum helps us connect the numbers. It allows us to use this relationship while solving the problem.

By establishing this first equation, we create a foundation upon which we can build further steps to solve for the unknowns.
Difference of Numbers
The exercise also involves the difference between two numbers, which is another key concept. According to the problem, the difference of the numbers is 9. This relationship can be expressed as:

  • x - y = 9
Again, x and y are our unknown numbers.
  • The difference helps us understand how much larger one number is compared to the other.
With both the sum and the difference equations, we have a system of linear equations:

  • x + y = 37
  • x - y = 9
This pair of equations lets us use various methods to find the values of x and y.
Substitution Method
To solve a system of linear equations, one of the most efficient methods is the substitution method. Here's how it works:

The goal is to isolate one variable and substitute it into the other equation, making it easier to find the values of both variables.

First, from our second equation:
  • x - y = 9
We can solve for x in terms of y:
  • x = y + 9
Next, substitute this expression for x back into the first equation:
  • (y + 9) + y = 37
Simplify and solve for y:

  • 2y + 9 = 37
  • 2y = 28
  • y = 14
Finally, substitute y = 14 back into one of the original equations to find x:
  • x + 14 = 37
  • x = 23
By following these steps, you'll find x and y easily and verify that they satisfy both original conditions.

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