Problem 253 - Elementary Algebra

Question

In the following exercises, translate to a system of equations and solve. Hannah has to make twentyfive gallons of punch for a potluck. The punch is made of soda and fruit drink. The cost of the soda is \(\$ 1.79\) per gallon and the cost of the fruit drink is \(\$ 2.49\) per gallon. Hannah's budget requires that the punch cost \(\$ 2.21\) per gallon. How many gallons of soda and how many gallons of fruit drink does she need?

Step-by-Step Solution

Answer

Hannah needs 10 gallons of soda and 15 gallons of fruit drink.

Step 1: Define the Variables

Let x represent the number of gallons of soda and y represent the number of gallons of fruit drink that Hannah needs to use.

Step 2: Set Up the Equation for Total Volume

Since Hannah needs to make a total of 25 gallons of punch, we have the equation: x + y = 25

Step 3: Set Up the Equation for Total Cost

The total cost of the punch should be \( 2.21 \) per gallon. Therefore, we have the cost equation: 1.79x + 2.49y = 2.21 * 25

Step 4: Simplify the Cost Equation

Simplify the second equation to get: 1.79x + 2.49y = 55.25

Step 5: Solve the System of Equations

First, solve the first equation for y to get: y = 25 - x Substitute this expression in place of y in the second equation: 1.79x + 2.49(25 - x) = 55.25

Step 6: Solve for x

Distribute the 2.49 in the equation: 1.79x + 62.25 - 2.49x = 55.25 Combine like terms: -0.70x + 62.25 = 55.25 Subtract 62.25 from both sides: -0.70x = -7 Divide by -0.70: x = 10

Step 7: Solve for y

Use the value of x to find y: y = 25 - x y = 25 - 10 y = 15

Step 8: Verify the Solution

Substitute x and y back into the original cost equation to check: 1.79(10) + 2.49(15) = 55.25 17.90 + 37.35 = 55.25 55.25 = 55.25

Key Concepts

Algebra

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating those symbols. In this exercise, algebra helps us represent unknown quantities (like the gallons of soda and fruit drink) using variables. For example, we let \(x\) represent the gallons of soda and \(y\) represent the gallons of fruit drink.

By setting up equations based on the problem's facts, we use algebra to solve for these unknowns. The variables and equations help us transform a word problem into a mathematical form that we can manipulate and solve.

To summarize, basic algebraic skills are essential for setting up and solving these types of problems.

Word Problems

Word problems translate real-world situations into mathematical statements. The key to solving word problems is to carefully read the problem, identify the unknowns, and express the relationships between the quantities using equations.

In this exercise, we are told about the total volume and cost of two types of beverages being mixed.

• We need to make 25 gallons of punch (total volume).
• The cost per gallon is \(2.21\ \text{dollars} \).
• The cost per gallon for soda and fruit drink are different.

These details help us shape our mathematical equations. Identifying keywords and relationships is critical.

Linear Equations

A linear equation is an equation between two variables that gives a straight line when plotted on a graph. In this exercise, we use linear equations to model the problem.

The first equation \(x + y = 25\) represents the total volume of punch. The second equation \(1.79x + 2.49y = 55.25\) is the cost equation simplified from \(2.21 \times 25 \). Linear equations are vital for connecting different quantities in an easily solvable form.

We solve systems of linear equations to find the values of \(x \) and \(y\). These solutions give us the number of gallons of each drink needed.

Solving Systems

Solving systems of equations involves finding the values of the variables that satisfy all equations in the system simultaneously. Here, we have two linear equations:

• \(x + y = 25\)
• \(1.79x + 2.49y = 55.25\)

One common method to solve these systems is substitution, which we used in this exercise. This involves:

• Solving one equation for one variable.
• Substituting that expression into the other equation.
• Simplifying to solve for the remaining variable.

Following this method, we determined \(x = 10\) and \(y = 15\), meaning 10 gallons of soda and 15 gallons of fruit drink meet Hannah's requirements.

Systems of equations enable us to handle multiple relationships at once, providing a powerful tool for complex problem-solving.