Problem 47 - Elementary Algebra

Question

Suppose that on a particular day the cost of 3 tennis balls and 2 golf balls is \(\$ 12\). The cost of 6 tennis balls and 3 golf balls is \(\$ 21\). Find the cost of 1 tennis ball and the cost of 1 golf ball.

Step-by-Step Solution

Answer

The cost of a tennis ball is $2, and a golf ball is $3.

Step 1: Translation into Equations

Let's represent the cost of one tennis ball as \(x\) and the cost of one golf ball as \(y\). The first scenario gives us the equation \(3x + 2y = 12\). The second scenario gives us the equation \(6x + 3y = 21\). Our goal is to find the values of \(x\) and \(y\).

Step 2: Simplification of Equations

First, simplify the second equation. Notice that both terms in \(6x + 3y = 21\) can be divided by 3, giving us \(2x + y = 7\). Now, we have the system of equations: \(3x + 2y = 12\) and \(2x + y = 7\).

Step 3: Substitution Method

Solve the simplified second equation for \(y\). From \(2x + y = 7\), we find \(y = 7 - 2x\). Substitute \(y = 7 - 2x\) into the first equation: \(3x + 2(7 - 2x) = 12\).

Step 4: Solving for x

Substitute \(y = 7 - 2x\) in the first equation, giving \(3x + 14 - 4x = 12\). Simplify the equation: \(-x + 14 = 12\). Solving this, we find \(-x = -2\), so \(x = 2\). The cost of one tennis ball is \(\$2\).

Step 5: Finding y

Now substitute \(x = 2\) into the expression for \(y\): \(y = 7 - 2(2)\). This simplifies to \(y = 3\). The cost of one golf ball is \(\$3\).

Key Concepts

systems of equations

In mathematics, when we talk about a 'system of equations', we are dealing with multiple equations that are interrelated and need to be solved together. In this particular exercise, we are given two equations derived from the cost of tennis and golf balls:

• The first equation is \(3x + 2y = 12\), representing the cost scenario with 3 tennis balls and 2 golf balls earning a total cost of \\(12.
• The second equation, \(6x + 3y = 21\), is related to the cost of 6 tennis balls and 3 golf balls totaling \\)21.

Each equation contains two unknowns: \(x\) (cost of a tennis ball) and \(y\) (cost of a golf ball). Together, these equations form a system that needs to be solved simultaneously. The solution to a system of equations is the set of values that satisfy all equations simultaneously. Thus, our goal is to find values for \(x\) and \(y\) that make both equations true. Understanding how to set up and solve systems of equations is crucial in algebra and can be applied to many practical situations.

substitution method

To solve a system of equations, one effective technique is the 'substitution method'. This method involves solving one of the equations for one variable and then substituting that expression into the other equation. Here's how it works step by step in the given exercise:

• Initially, we decided to simplify the second equation \(6x + 3y = 21\) by dividing each term by 3, resulting in \(2x + y = 7\).
• The next step involves isolating one variable. From \(2x + y = 7\), we solve for \(y\), giving us \(y = 7 - 2x\).
• Now, we take this expression for \(y\) and substitute it into the first equation \(3x + 2y = 12\). This yields the equation: \(3x + 2(7 - 2x) = 12\).

This method simplifies the process, as it reduces the systems of equations from two variables to a single equation with one variable. This allows us to solve for \(x\) first, turning the complex multi-variable problem into a more manageable single-variable problem.

equation simplification

Equation simplification is a critical skill in solving algebra problems efficiently. In this exercise, simplification plays a key role. Let's break down how this works and why it is useful:

• We started with the equation \(6x + 3y = 21\), which seems somewhat complex initially. However, by observing that each term is divisible by 3, we can simplify the equation to \(2x + y = 7\).
• This step not only makes the equation easier to work with but also helps in finding solutions faster. Simpler equations reduce calculation errors and improve your problem-solving speed.
• In the substitution method, we saw that simplifying expressions further, such as calculating \(y = 7 - 2x\), allows us to inject this into the first equation directly, reducing unnecessary steps.

Simplifying equations streamlines the overall process, saves time, and eliminates the chances of making mistakes in multiple calculation steps. It's a powerful technique applied frequently in algebra and beyond.