Question

Last season, a soccer team won five times as many games as it lost, with 8 games ending in a draw. If there were 38 games in the season, how many games did the team win? (A) 5 (B) 6 (C) 24 (D) 25

Short answer

The soccer team won 25 games in the season. The correct answer is (D) 25.

Step-by-step solution

Define the variables

Let W represent the number of games won, L represent the number of games lost, and D represent the number of games that ended in a draw. We are given the following information: 1. W = 5L 2. W + L + D = 38 3. D = 8

Substitute known values into the equation

We know that D = 8, so we can substitute this into the second equation: W + L + 8 = 38

Simplify the equation

Subtract 8 from both sides of the equation to isolate W and L: W + L = 30

Substitute the first equation into the simplified equation

We know that W = 5L, so we can substitute this into the equation W + L = 30: 5L + L = 30

Solve for L

Combine the like terms on the left side of the equation: 6L = 30 Now, divide both sides by 6 to get the value of L: L = 5

Find the value of W

Since W = 5L, we can substitute the value of L into the equation to find the number of games won: W = 5 × 5 = 25 The soccer team won 25 games in the season. The correct answer is (D) 25.

Key concepts

Linear Equations

Linear equations are fundamental in understanding algebra. They are called 'linear' because when you graph them, they form a straight line. A linear equation represents a balance, similar to a seesaw where what you do on one side must be done on the other to keep it level. In the soccer team exercise, the equations were derived based on real-world conditions: games won, lost, and drawn must add up to the total number of games played. This relationship forms a linear equation.

• An equation like \(W + L + D = 38\) indicates that the total outcome from all games must be equal to 38.
• Another equation, \(W = 5L\), shows a specific relationship between games won and lost: for every game lost, five were won.

The key to handling these equations is recognizing how they express conditions and constraints in a problem. They help us find an unknown value, using known quantities and relationships.

Variables in Algebra

Variables in algebra are symbols or letters that represent unknown numbers. In this context, they are used to set up equations that model real-world problems. In the soccer team scenario, the variables were:

• \(W\) for games won
• \(L\) for games lost
• \(D\) for games drawn

These variables allow you to create equations without needing to know the actual numbers beforehand. You assign each unknown quantity a variable and then use logical reasoning and mathematical operations to find their values. Understanding variables is crucial because it allows flexibility and generalization in solving problems, which are essential skills in mathematics.

Step-by-step Solution

Step-by-step solutions help break down complex problems into manageable parts, making them easier to understand and solve. For the soccer team problem, each step logically follows the previous one:

• First, identify and define all necessary variables and equations based on the problem description.
• Second, substitute known values into the equations to simplify them.
• Then, manipulate the equations to isolate variables, such as subtracting or dividing on both sides.
• Finally, use the simplified equations to find the unknown values.

This methodical approach ensures that no detail is overlooked, and each part of the problem is addressed systematically. By following these steps, students can strengthen their problem-solving skills and confidence in tackling various math challenges.