Problem 51 - Intermediate Algebra

Question

Suppose that on a particular day the cost of 3 tennis balls and 2 golf balls is $$\$ 7$$. The cost of 6 tennis balls and 3 golf balls is $$\$ 12$$. Find the cost of 1 tennis ball and the cost of 1 golf ball.

Step-by-Step Solution

Answer

One tennis ball costs $1 and one golf ball costs $2.

Step 1: Define the Variables

Let's denote the cost of one tennis ball as \( x \) and the cost of one golf ball as \( y \). We need to find the values of \( x \) and \( y \).

Step 2: Formulate the Equations

Based on the given statements, we can form the following system of equations: \( 3x + 2y = 7 \) (for 3 tennis balls and 2 golf balls) and \( 6x + 3y = 12 \) (for 6 tennis balls and 3 golf balls).

Step 3: Simplify the Second Equation

The second equation can be simplified by dividing all terms by 3: \( \frac{6x}{3} + \frac{3y}{3} = \frac{12}{3} \), which simplifies to \( 2x + y = 4 \).

Step 4: Use Substitution or Elimination

We can use the equation \( 2x + y = 4 \) to express \( y \) in terms of \( x \): \( y = 4 - 2x \).

Step 5: Substitute into the First Equation

Substitute \( y = 4 - 2x \) into the first equation: \( 3x + 2(4 - 2x) = 7 \).

Step 6: Solve for x

Simplify and solve the equation: \( 3x + 8 - 4x = 7 \)\( -x + 8 = 7 \)\( -x = -1 \)\( x = 1 \).

Step 7: Solve for y

Substitute \( x = 1 \) back into the equation \( y = 4 - 2x \): \( y = 4 - 2(1) \)\( y = 4 - 2 \)\( y = 2 \).

Key Concepts

Algebra

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating these symbols. In algebra, we often deal with equations and the unknown values they contain. These unknown values are typically represented by letters such as \( x \) and \( y \). The primary goal is to find the specific values of these unknowns that make the equation true.

In the context of this exercise, we are working with two main variables: the cost of a tennis ball, represented as \( x \), and the cost of a golf ball, represented as \( y \). Each equation formed from the problem's scenario contains these variables, showing how their linear combination results in a specific cost. Understanding algebra helps in the development of critical thinking skills by teaching us how to manipulate these symbols and solve for unknowns in practical situations.

Overall, algebra is not just about numbers; it's about recognizing patterns and relationships, applying logical reasoning, and solving problems efficiently.

Linear Equations

Linear equations are equations of the first order, which means they involve no higher powers than the first power of the variable involved. Structurally, a linear equation can be written in the form \( ax + by = c \), where \( a \), \( b \), and \( c \) are constants, and \( x \) and \( y \) are variables.

In this problem, we have a system of linear equations:

• \(3x + 2y = 7\)
• \(2x + y = 4\)

These linear equations represent a relationship between the number of tennis and golf balls and their respective costs. Each equation graphically represents a straight line on a coordinate plane, and the solution to the system of equations is the point where these lines intersect.

Solving linear equations is crucial because it allows us to determine unknown quantities in various real-world and mathematical contexts quickly and accurately.

Variable Substitution

Variable substitution is a powerful method used to solve systems of equations. In this technique, one equation is manipulated to express one variable in terms of the other variable(s). Then, this expression is substituted into another equation to solve for one of the variables.

In the exercise, substitution was used effectively by expressing \( y \) in terms of \( x \) from the simplified equation \( 2x + y = 4 \). This gives us \( y = 4 - 2x \). By substituting this expression into the first equation \( 3x + 2y = 7 \), we could simplify the problem to finding the value of \( x \).

The step-by-step substitution looks like this:

• Substitute \( y = 4 - 2x \) into \( 3x + 2y = 7 \), resulting in \( 3x + 2(4 - 2x) = 7 \)
• Simplify and solve for \( x \) and then \( y \).

This method is particularly helpful when dealing with complex systems, as it reduces the number of equations and makes the process of finding solutions more manageable.