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Magazine Chapter 8: Problem 45
Find two numbers that have a difference of 8 and a quotient of 3.
Short Answer
Expert verified
The two numbers are 12 and 4.
Step by step solution
01
Define the Variables
Let the two numbers be represented by \( x \) and \( y \). We will set up two equations based on the information given in the problem statement. Let \( x > y \).
02
Set Up the Difference Equation
The problem states that the difference of the two numbers is 8. Therefore, we write the equation \( x - y = 8 \).
03
Set Up the Quotient Equation
The problem also gives us that the quotient of the two numbers is 3, which can be written as \( \frac{x}{y} = 3 \).
04
Express x in Terms of y Using the Quotient Equation
From the equation \( \frac{x}{y} = 3 \), multiply both sides by \( y \) to get \( x = 3y \).
05
Substitute Expression into Difference Equation
Substitute \( x = 3y \) into the difference equation \( x - y = 8 \). This gives us \( 3y - y = 8 \).
06
Simplify and Solve for y
Simplify the equation to get \( 2y = 8 \). Divide both sides by 2 to find \( y = 4 \).
07
Use the Value of y to Find x
Now that we have \( y = 4 \), substitute back into \( x = 3y \) to find \( x = 3(4) = 12 \).
08
Verify the Solution
Check that the solution satisfies both original conditions: The difference is \( x - y = 12 - 4 = 8 \) and the quotient is \( \frac{x}{y} = \frac{12}{4} = 3 \). Both conditions are satisfied.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Difference Equation
A difference equation helps us understand how two numbers relate by their subtraction. In the context of this exercise, we have two unknown numbers, which we call \( x \) and \( y \).
The equation for this difference is \( x - y = 8 \), meaning that when we subtract the smaller number, \( y \), from the larger number, \( x \), we should get 8.
This is a crucial step in forming a system of equations because it provides one of the necessary relationships between our variables.
The equation for this difference is \( x - y = 8 \), meaning that when we subtract the smaller number, \( y \), from the larger number, \( x \), we should get 8.
This is a crucial step in forming a system of equations because it provides one of the necessary relationships between our variables.
- This equation is linear, meaning it forms a straight line when graphed.
- You need this equation to represent how much larger the number \( x \) is compared to \( y \).
Quotient Equation
The quotient equation relates two numbers through division. In this problem, the quotient of the two numbers is 3, which is expressed as \( \frac{x}{y} = 3 \).
We know that dividing \( x \) by \( y \) should yield 3, showing that \( x \) is three times as large as \( y \).
This equation helps us further narrow down the values of \( x \) and \( y \), creating a system of equations together with the difference equation.
We know that dividing \( x \) by \( y \) should yield 3, showing that \( x \) is three times as large as \( y \).
This equation helps us further narrow down the values of \( x \) and \( y \), creating a system of equations together with the difference equation.
- The quotient equation is also an expression of direct proportion between \( x \) and \( y \).
- From this equation, \( x \) can be expressed in terms of \( y \) as \( x = 3y \).
Solve for Variables
To solve for variables in a system of equations, such as in this exercise, substitution is an effective method. After expressing \( x \) in terms of \( y \) as \( x = 3y \) from the quotient equation, we substitute this expression into the difference equation.
The substitution transforms \( x - y = 8 \) into \( 3y - y = 8 \).
Simplifying this equation gives \( 2y = 8 \). Divide by 2 to find \( y = 4 \).
Once you have \( y \), use it to find \( x \) by substituting back into \( x = 3y \), resulting in \( x = 12 \).
The substitution transforms \( x - y = 8 \) into \( 3y - y = 8 \).
Simplifying this equation gives \( 2y = 8 \). Divide by 2 to find \( y = 4 \).
Once you have \( y \), use it to find \( x \) by substituting back into \( x = 3y \), resulting in \( x = 12 \).
- This step involves basic algebraic manipulation: substitution, simplification, and solving linear equations.
- Always double-check your substitute values to ensure accuracy.
Verify Solution
Verifying a solution is an essential final step in solving a system of equations. This step ensures that the calculated values satisfy all given conditions of the problem.
For our found values \( x = 12 \) and \( y = 4 \), we need to check:
Verification is crucial because it identifies and corrects potential errors, ensuring that the solution makes sense in the context of the original problem.
For our found values \( x = 12 \) and \( y = 4 \), we need to check:
- Difference: \( x - y = 12 - 4 = 8 \) 鈥� It matches the given condition.
- Quotient: \( \frac{x}{y} = \frac{12}{4} = 3 \) 鈥� Also satisfies the condition.
Verification is crucial because it identifies and corrects potential errors, ensuring that the solution makes sense in the context of the original problem.
