Problem 36 - Algebra and Trigonometry

Question

Use the four-step strategy to solve each problem. Use \(x, y,\) and \(z\) to represent unknown quantities. Then translate from the verbal conditions of the problem to a system of three equations in three variables. On a recent trip to the convenience store, you picked up 1 gallon of milk, 7 bottles of water, and 4 snack-size bags of chips. Your total bill (before tax) was $$ 17.00 .\( If a bottle of water costs twice as much as a bag of chips, and a gallon of milk costs \)\$ 2.00$ more than a bottle of water, how much does each item cost?

Step-by-Step Solution

Answer

The cost of each item is: a gallon of milk costs $3.50, a bottle of water costs $1.50, and a bag of chips costs $0.75.

Step 1: Identify the variables

First, name the variables. Let \(x\) represent the cost of a gallon of milk, \(y\) the cost of a bottle of water, and \(z\) the cost of a bag of chips.

Step 2: Translate the problem into system of equations

Now, translate the problem conditions into a system of equations:- The total cost for 1 gallon of milk, 7 bottles of water, and 4 bags of chips is $17.00. So, this leads to the equation: \(x + 7y + 4z = 17\).- A bottle of water (\(y\)) costs twice as much as a bag of chips (\(z\)). So, this can be written as: \(y = 2z\).- A gallon of milk (\(x\)) costs $2.00 more than a bottle of water (\(y\)). So, this can be written as: \(x = y + 2\).

Step 3: Substitute the equations

Substitute the equations in order to make them easier to solve. Substitute \(y\) from the second equation into the first and third: \(x + 7(2z) + 4z = 17\) and \(x = 2z + 2\). Simplify the equations to: \(x + 18z = 17\) and \(x = 2z + 2\).

Step 4: Solve the equations

Now, set the two equations \(x + 18z = 17\) and \(x = 2z + 2\) equal to each other since both are equal to \(x\). That gives us \(2z + 2 + 18z = 17\). Simplify and solve for \(z\): \(20z = 15\) so \(z = 15 / 20 = 0.75\). Now that we know \(z\), we can substitute it back into the equation \(y = 2z\) to find \(y = 2 * 0.75 = 1.5\). Lastly, substitute \(y = 1.5\) into the equation \(x = y + 2\) to find \(x = 1.5 + 2 = 3.5\).

Step 5: Interpret the solution

The solution shows that a gallon of milk (represented by \(x\)) costs $3.50, a bottle of water (represented by \(y\)) costs $1.50, and a bag of chips (represented by \(z\)) costs $0.75.

Key Concepts

Variable Substitution

Variable substitution is a fundamental step in solving systems of equations, especially when dealing with problems involving multiple variables. In this context, variable substitution involves replacing one variable with its equivalent expression in terms of another variable.
This technique simplifies the equations and makes them more manageable. For example, in the given problem, we start with identifying three variables:

• Letting \( x \) represent the cost of a gallon of milk,
• \( y \) the cost of a bottle of water,
• and \( z \) the cost of a bag of chips.

By knowing that \( y = 2z \) (since a bottle of water costs twice as much as a bag of chips), and that \( x = y + 2 \) (a gallon of milk costs $2 more than water), we substitute these relationships into the other equations to eliminate variables and reduce the complexity of solving the system.

Linear Equations

Linear equations are mathematical statements that depict the relationship between variables using a straight line when graphed. These equations are essential in solving real-world problems like the current exercise.
The problem interpretation yields three linear equations:

• \( x + 7y + 4z = 17 \), representing the total cost of items,
• \( y = 2z \), expressing the relationship between the cost of water and chips,
• and \( x = y + 2 \), expressing the cost of milk in relation to water.

Working with linear equations allows us to analyze relationships between variables and easily find solutions using algebraic methods. In this case, graphing these equations would provide intersection points which correspond to the solution, but using algebraic manipulation is often more efficient.

Verbal Conditions

Translating verbal conditions into mathematical equations is a critical skill in problem solving. These verbal conditions form the foundation upon which the mathematical model is built.
To tackle this problem, it's essential to clearly understand the descriptions given in words:

• "A bottle of water costs twice as much as a bag of chips." This gives us the equation \( y = 2z \).
• "A gallon of milk costs \(2 more than a bottle of water." This gives us \( x = y + 2 \).
• Lastly, the cost of all items totals \)17. This is expressed as \( x + 7y + 4z = 17 \).

These verbal conditions are translated into linear equations, laying the groundwork for finding a mathematical solution.

Problem Solving

The ultimate goal of problem solving in the context of systems of equations is to find values for the variables that satisfy all the given equations simultaneously. This often requires a well-organized strategy:

• Identify and rename variables (e.g., \( x, y, \) and \( z \)).
• Translate the situation into a system of equations (as done from verbal conditions).
• Use techniques like substitution to simplify and solve the system.
• Finally, interpret the derived solution in the context of the original problem.

The step-by-step solution of this exercise demonstrated how these strategies converge to give meaningful answers: the cost of milk, water, and chips. Effective problem solving requires not only the determination of mathematical solutions but also ensuring they align with real-world interpretations, like identifying incorrect or improbable values.