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Magazine Chapter 5: Problem 40
Solve the given problems by determinants. In Exerciser-46,\( set up appropriate systems of equations. All numbers are accurate to at least two significant digits. A certain 18 -hole golf course has par- \)3,\( par- \)4,\( and par- 5 holes, and there are twice as many par- 4 holes as par- 5 holes. How many holes of each type are there if a golfer has par on every hole for a score of \)70 ?$
Short Answer
Expert verified
There are 6 par-3 holes, 8 par-4 holes, and 4 par-5 holes.
Step by step solution
01
Defining Variables
Let's define the variables to represent the number of par-3, par-4, and par-5 holes. Let \( x \) be the number of par-3 holes, \( y \) be the number of par-4 holes, and \( z \) be the number of par-5 holes.
02
Setting Up Equations Based on Hole Types
According to the problem, the total number of holes is 18. Therefore, we have the equation:\[ x + y + z = 18 \]
03
Setting Up Equations Based on Par Scores
The golfer's total score of 70 comes from scoring par on each hole. This gives us the equation:\[ 3x + 4y + 5z = 70 \]
04
Relationship Between Par-4 and Par-5 Holes
The problem states that there are twice as many par-4 holes as par-5 holes. This relationship provides the equation:\[ y = 2z \]
05
Substituting the Relationship into the Equations
Substitute \( y = 2z \) into the first two equations. The equations become:1. \[ x + 2z + z = 18 \] which simplifies to \[ x + 3z = 18 \] 2. \[ 3x + 4(2z) + 5z = 70 \] simplifies to \[ 3x + 8z + 5z = 70 \] or \[ 3x + 13z = 70 \]
06
Solve the System of Equations
We now have two equations to solve:1. \( x + 3z = 18 \) 2. \( 3x + 13z = 70 \) Use the substitution method:- From equation 1, express \( x \): \[ x = 18 - 3z \] - Substitute \( x \) in the second equation: \[ 3(18 - 3z) + 13z = 70 \] \[ 54 - 9z + 13z = 70 \] \[ 4z = 16 \] \[ z = 4 \]
07
Solve for Remaining Variables
With \( z = 4 \), substitute back to find \( x \) and \( y \):- \( x = 18 - 3(4) = 18 - 12 = 6 \) - \( y = 2z = 2(4) = 8 \)
08
Conclusion on Number of Holes by Type
The golf course has 6 par-3 holes, 8 par-4 holes, and 4 par-5 holes. These values satisfy all the conditions of the problem.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
systems of equations
In mathematics, a system of equations is a set of two or more equations involving a number of variables. Think of each equation as a rule or a condition that the variables must meet. The objective is to find values for the variables that satisfy all the rules simultaneously. In the context of the golf course problem, we have three types of holes, each requiring an appropriate variable assignment:
- Let \( x \) represent the number of par-3 holes.
- Let \( y \) represent the number of par-4 holes.
- Let \( z \) represent the number of par-5 holes.
- The total number of holes is 18: \( x + y + z = 18 \).
- The golfer's total score is a par of 70: \( 3x + 4y + 5z = 70 \).
- There are twice as many par-4 holes as par-5 holes: \( y = 2z \).
substitution method
The substitution method is a common and effective way to solve systems of equations. It's like a puzzle where you work step-by-step to replace variables with things you already know or can calculate. Here's how we used the substitution method for the golf course problem:First, we took one of the equations which expresses a variable in terms of another, such as \( y = 2z \), to express \( y \). Step-by-Step:
- Begin by substituting \( y = 2z \) in the equations:
- Now the first equation becomes \( x + 3z = 18 \).
- The second equation becomes: \( 3x + 13z = 70 \).
- By simplifying, you can solve one equation for one variable, e.g., \( x = 18 - 3z \).
- Substitute \( x = 18 - 3z \) back in the remaining equation.
linear algebra
Linear algebra is a branch of mathematics that deals with vectors, matrices, and systems of linear equations.
In problems like our golf course scenario, linear algebra offers structured strategies to manage equations systematically.
At its heart, linear algebra allows us to approach complex systems using concepts like:
- Equations as Lines: Each equation in our system represents a line (or plane in higher dimensions). The solution is where these lines intersect.
- Matrix Representation: Though not used explicitly here, systems of equations can be expressed in matrix form, making them easier to manipulate.
- Determinants and Inverses: Useful in solving systems, though simple substitution was sufficient here.
golf course problem
This particular golf course problem is an interesting exercise in applying mathematical reasoning to a real-world scenario.
It combines logical thinking with systems of equations to solve for the composition of a golf course.
Breaking Down the Problem:
- Total Holes: We know there are 18 holes in total, a mix of par-3, par-4, and par-5 holes.
- Total Score: A golfer scores par on every hole for a total of 70. This gives us a clue into how these scores spread out across the different types of holes.
- Relative Quantities: The puzzle adds an extra layer by stating there are twice as many par-4 holes as there are par-5 holes.
