Problem 40 - College Algebra with Modeling and Visualization

Question

Business Production A business has three machines that manufacture containers. Together they can make 100 containers per day, whereas the two fastest machines can make 80 containers per day. The fastest machine makes 34 more containers per day than the slowest machine. (a) Let \(x, y,\) and \(z\) be the numbers of containers that the machines make from fastest to slowest. Write a system of three equations whose solution gives the number of containers each machine can make. (b) Solve the system of equations.

Step-by-Step Solution

Answer

The machines make 54, 26, and 20 containers per day, respectively.

Step 1: Define Variables

Let the number of containers made by the fastest machine be \(x\), the second fastest machine be \(y\), and the slowest machine be \(z\). We need to write the corresponding equations based on the problem statement.

Step 2: Create the First Equation

All three machines together can produce 100 containers in a day. Therefore, the first equation based on the total production is: \[ x + y + z = 100 \]

Step 3: Create the Second Equation

The two fastest machines produce 80 containers together. So, the equation based on their production is: \[ x + y = 80 \]

Step 4: Create the Third Equation

The fastest machine makes 34 more containers than the slowest machine. Thus, the third equation is: \[ x = z + 34 \]

Step 5: Substitute and Solve the Equations

1. Substitute \(x = z + 34\) into the equations. Start with \(x + y + z = 100\): \((z + 34) + y + z = 100\) Simplify to: \(y + 2z = 66\).2. Substitute \(x = 80 - y\) into the expression of \(x = z + 34\). \(80 - y = z + 34\) Simplify to: \(z = 46 - y\).3. Substitute \(z = 46 - y\) into \(y + 2z = 66\): \(y + 2(46 - y) = 66\) Simplify to: \(y + 92 - 2y = 66\) \(-y + 92 = 66\) \(-y = 66 - 92\) \(y = 26\).4. Substitute \(y = 26\) back into \(x = 80 - y\): \(x = 80 - 26\) \(x = 54\).5. Substitute \(y = 26\) into \(z = 46 - y\): \(z = 46 - 26\) \(z = 20\).

Step 6: Verify the Solutions

Check if the solutions satisfy all original equations:1. \(x + y + z = 54 + 26 + 20 = 100\).2. \(x + y = 54 + 26 = 80\).3. \(x = z + 34 = 20 + 34 = 54\).All equations are satisfied, confirming our solutions.

Key Concepts

Linear Equations

Linear equations form the backbone of algebraic problem-solving. They are mathematical statements expressing relationships using letters known as variables, which stand in for unknown values.
In this specific exercise, three linear equations are formed to represent the production of containers by three machines. Each equation captures a different aspect of the problem:

• The total production by all three machines is expressed as \( x + y + z = 100 \).
• The production by the fastest two machines is represented by \( x + y = 80 \).
• The relationship between the fastest and the slowest machine is shown as \( x = z + 34 \).

These equations highlight how linear equations can capture different real-world scenarios through mathematical relationships.
Understanding them helps break down complex problems into simpler components.

Algebraic Problem-Solving

Solving algebraic problems involves finding the values of the unknown variables in a set of equations. This is done by manipulating the equations according to algebraic rules to isolate and solve for each variable.
In algebraic problem-solving, the exercise outlines clear steps to derive the solution. Each equation provides a piece of the puzzle; collectively, they tell us about the production capabilities of each machine in the system.
Here's what you can do to tackle similar problems:

• Define your variables clearly. This makes it easier to create equations from the problem context.
• Write equations based on the relationships given in the problem statements.
• Systematically manipulate the equations to simplify and solve them, often working from one equation to another.

The goal is to logically deduce the unknowns, thereby solving the problem.

Substitution Method

The substitution method is a powerful tool for solving systems of equations. It involves solving one equation for a variable and then substituting that expression into another equation. This reduces the number of variables and simplifies the problem.
In the exercise example, we can use the substitution method to gradually find each machine's production:

• The equation \( x = z + 34 \) gives \( x \) in terms of \( z \). By substituting this into \( x + y + z = 100 \), we simplify it to \( y + 2z = 66 \).
• Similarly, from \( x + y = 80 \), we isolate \( x \) as \( x = 80 - y \) and use this to further simplify the equations.
• Finally, we replace \( z = 46 - y \) into the last equation, allowing us to solve for \( y \) and subsequently find \( x \) and \( z \).

The substitution method effectively reduces confusion, allowing you to work with one variable at a time until all the unknowns are found and verified.