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To solve a problem involving a golf course's par score, we need to understand how different holes contribute to the total par score. A golf course typically has three types of holes: par-3, par-4, and par-5 holes. Each type of hole contributes differently to the total par score based on the number of strokes assigned:

• Par-3 holes contribute 3 strokes each.
• Par-4 holes contribute 4 strokes each.
• Par-5 holes contribute 5 strokes each.

In a par score calculation, the total strokes from all holes sum up to the player's overall score when they achieve par on each hole. In the given exercise, the total par score is 70 for all 18 holes. By setting up an equation that sums up each hole's contribution, we can determine the number of each type of hole. This is achieved by creating an equation that represents the total par: \[ 3x + 4y + 5z = 70 \]This equation ensures that each hole type's par is accounted for, aiding us in finding how many holes there are of each type on the course.

When solving a system of equations, defining variables is a crucial first step. Each variable represents a quantity we need to determine. In our golf course example:

• Let \( x \) be the number of par-3 holes.
• Let \( y \) be the number of par-4 holes.
• Let \( z \) be the number of par-5 holes.

The variables help us create equations that reflect the relationships between the different hole types and the given conditions of the problem. In our problem, these relationships include:

• The total number of holes: \( x + y + z = 18 \)
• The relationship between par-4 and par-5 holes: \( y = 2z \)
• The par score calculation combining all hole types discussed earlier.

By defining and utilizing these variables, we can clearly understand and organize the information given in the problem, setting up a manageable system of equations for solving the exercise.

Solving a system of equations involves a structured process: using algebraic methods to find values for the variables. In our golf exercise, after establishing our equations, we proceed step by step:**Step 1: Simplify and Substitute**
First, we use the relationship \( y = 2z \) to substitute for \( y \) in our equations, transforming equation 1 to \( x + 3z = 18 \) and equation 2 to \( 3x + 13z = 70 \).**Step 2: Solve for One Variable**
We subtract the first transformed equation from the second to isolate variables, simplifying it to \( 2x + 10z = 52 \).**Step 3: Isolate \( z \)**
Dividing by 2, we get \( x + 5z = 26 \). Then subtract \( x + 3z = 18 \) from this, leaving us with \( 2z = 8 \), giving \( z = 4 \).**Step 4: Solve for Remaining Variables (\( x \) and \( y \))**
Substitute \( z \) into \( y = 2z \), getting \( y = 8 \). Then substitute \( z = 4 \) into the equation \( x + 3z = 18 \) to find \( x = 6 \).This careful step-by-step process allows us to find the solution: 6 par-3 holes, 8 par-4 holes, and 4 par-5 holes, ensuring that we meet all conditions set by the problem.