Problem 58 - College Algebra

Question

For the following exercises, set up the augmented matrix that describes the situation, and solve for the desired solution. The three most popular ice cream flavors are chocolate, strawberry, and vanilla, comprising \(83 \%\) of the flavors sold at an ice cream shop. If vanilla sells \(1 \%\) more than twice strawberry, and chocolate sells \(11 \%\) more than vanilla, how much of the total ice cream consumption are the vanilla, chocolate, and strawberry flavors?

Step-by-Step Solution

Answer

Vanilla: 29%, Chocolate: 40%, Strawberry: 14%.

Step 1: Define Variables

Let the percentages of vanilla, chocolate, and strawberry sold be \( v \), \( c \), and \( s \) respectively. These variables will represent the relative amounts sold of each flavor.

Step 2: Set Up Equations

Based on the problem statement, we have the following equations:1. The total percentage of vanilla, chocolate, and strawberry is 83%: \[ v + c + s = 83 \]2. Vanilla sells 1% more than twice strawberry: \[ v = 2s + 1 \]3. Chocolate sells 11% more than vanilla: \[ c = v + 11 \]

Step 3: Substitute and Simplify

Replace \( v \) and \( c \) in the first equation using the expressions from steps 2: - Substitute \( v = 2s + 1 \) into \( c = v + 11 \): \[ c = (2s + 1) + 11 = 2s + 12 \] - Substitute \( v = 2s + 1 \) and \( c = 2s + 12 \) into \( v + c + s = 83 \): \[ (2s + 1) + (2s + 12) + s = 83 \]

Step 4: Solve for Strawberry (s)

Combine like terms in the equation from Step 3: \[ 2s + 1 + 2s + 12 + s = 83 \] \[ 5s + 13 = 83 \]Subtract 13 from both sides: \[ 5s = 70 \]Divide by 5: \[ s = 14 \]

Step 5: Solve for Vanilla (v)

Use \( s = 14 \) in the equation \( v = 2s + 1 \): \[ v = 2(14) + 1 \] \[ v = 28 + 1 = 29 \]

Step 6: Solve for Chocolate (c)

Use \( v = 29 \) in the equation \( c = v + 11 \): \[ c = 29 + 11 \] \[ c = 40 \]

Step 7: Solution Verification

Verify the total percent: \( v + c + s = 29 + 40 + 14 = 83 \), which matches the given total. All conditions are satisfied.

Key Concepts

Systems of equations

A system of equations is a collection of two or more equations with a common set of unknowns. In this problem, the unknowns are the percentages of vanilla, chocolate, and strawberry ice cream sold. When we talk about systems of equations, these can be linear or non-linear.

• Linear Systems: All of the equations involved are first-degree polynomials.
• Non-linear Systems: At least one of the equations is a polynomial of degree greater than one or involves complex operations such as squares or cubes.

In this exercise, we are dealing with a linear system because the relationships between vanilla, chocolate, and strawberry follow linear equations. A solution to a system of equations is a set of values that make all equations true simultaneously. By formulating and solving these equations, we aim to determine the exact percentages of each ice cream flavor sold in this problem.

Substitution method

The substitution method is a well-known technique for solving systems of equations. It involves expressing one variable in terms of the other variable(s) and substituting this expression into the other equation(s). Here’s how it works:

• Start by choosing an equation and solve for one variable in terms of the others.
• Substitute this expression into the other equations. This reduces the number of variables in those equations.
• Continue the process until you can solve for each variable.

In our scenario, we begin by solving the equations for vanilla and chocolate in terms of the strawberry variable (\(v = 2s + 1\) and \(c = v + 11\)). Substituting these into the original system (\(v + c + s = 83\)), simplifies it and allows finding one single variable (\(s\)). This simplification step reduces the complexity of the system, making it much easier to solve.

Matrix algebra

Matrix algebra is a powerful mathematical tool that is prominently used for handling and solving systems of linear equations. An **augmented matrix** is often used in this context. It includes a matrix representation of the coefficients of the variables along with the constants from each equation.

• The rows represent each of the equations in the system.
• The columns represent each of the coefficients of the variables, with an additional column for the constants on the right-hand side.

Forming an augmented matrix from a system of equations makes it easier to use row operations and other methods to solve for the unknown variables. While our example used substitution, systems like this can be solved using matrix operations such as Row Echelon Form (REF) or Gaussian elimination, making matrix algebra a crucial tool in tackling larger and more complex systems of equations efficiently.

Solving linear equations

Solving linear equations involves finding the values of the unknown variables that satisfy all the equations in a system. In linear equations, all terms are either constants or the product of a constant and a single variable.

• The goal is to isolate the variables on one side of the equation, simplifying the problem to a point where the solution becomes evident.
• Typically, this involves manipulating the equations through addition, subtraction, multiplication, or division.

In the ice cream problem, we simplified the linear equations through a series of substitutions and simple arithmetic operations, ending in a clear calculation of each variable (\(s\), \(v\), and \(c\)). Verifying the solution by plugging back into the original equations confirmed that the calculated values satisfy all given conditions. This step is a necessary part of solving linear equations to ensure accuracy and correctness.