Problem 31 - Beginning and Intermediate Algebra

Question

In New York City, orange juice, a raisin bagel, and a cup of coffee from Katie's Koffee Kart cost a total of \(\$ 8.15 .\) Katie posts a notice announcing that, effective the following week, the price of orange juice will increase \(25 \%\) and the price of bagels will increase \(20 \% .\) After the increase, the same purchase will cost a total of \(\$ 9.30\), and the raisin bagel will cost \(30 €\) more than coffee. Find the price of each item before the increase.

Step-by-Step Solution

Answer

The initial price of the orange juice, raisin bagel, and cup of coffee were approximately \( \$2.95 \), \( \$2.50 \), and \( \$2.70 \), respectively.

Step 1: Solve the equation for one variable

We have the equation: \(1.20y = 1z + 0.30\) Let us isolate z: \(z = 1.20y - 0.30\) Now we can substitute this expression for \(z\) in the other equations to eliminate \(z\).

Step 2: Substitute into the equations and solve

Now substitute the expression for \(z\) in the other equations: - Initial total cost: \(x + y + (1.20y - 0.30) = 8.15\) - New total cost: \(1.25x + 1.20y + (1.20y - 0.30) = 9.30\) Simplify the equations: - Initial total cost: \(x + 2.20y - 0.30 = 8.15\) - New total cost: \(1.25x + 2.40y - 0.30 = 9.30\) Now we have two linear equations with two variables. We can solve the system of equations using, for example, the substitution method.

Step 3: Solve the system of equations

Let's solve the initial total cost equation for \(x\): \(x = 8.15 - 2.20y + 0.30\) Now substitute this expression for \(x\) in the new total cost equation: \(1.25 (8.15 - 2.20y + 0.30) + 2.40y - 0.30 = 9.30\) Now, we need to solve for \(y\): \(10.18 - 2.75y + 2.40y = 9.30\) \(0.35y = 0.88\) \(y = 2.51\approx 2.50\) Now that we have found the value of \(y\), we can find the value of \(x\) and soon after the value of \(z\): \(x = 8.15 - 2.20y + 0.30 = 8.15 - 2.20(2.51) + 0.30\) \(x = 8.15 - 5.52 + 0.30 = 2.93 \approx 2.95\) \(z = 1.20y - 0.30 = 1.20(2.51) - 0.30\) \(z = 3.01 - 0.30 = 2.71 \approx 2.70\) The initial price of orange juice (x) is approximately \(2.95, the initial price of a raisin bagel (y) is approximately \)2.50, and the initial price of coffee (z) is approximately $2.70.

Key Concepts

Understanding Systems of Equations

In mathematics, a system of equations refers to a set of equations with multiple variables that can be solved together. This means you're trying to find values for the variables that will satisfy all the equations at the same time. In many everyday situations, like calculating costs based on various factors, systems of equations prove to be useful.
Here's a simple way to look at it:

• Each equation gives you some information about the relationship between the variables involved.
• By solving the system, you find a solution that works for all the equations simultaneously.

In our problem, the system comprises two equations with varying costs, showcasing their simultaneous conditions before and after a price change, with each component (juice, bagel, coffee) representing a variable.

Applying the Substitution Method

The substitution method is an effective technique for solving a system of equations. It involves solving one of the equations for one variable, then substituting this expression into the other equations. This reduces the number of variables, simplifying the problem.
Here's how the substitution method works:

• Solve one equation for one variable. For instance, solve for \(z\) in terms of \(y\): \(z = 1.20y - 0.30\).
• Substitute this expression into the other equations. This step removes \(z\) from the equations, making it easier to find the value of \(y\).

Once you find one variable's value, use it to calculate the others. The substitution method helps break down complex algebra problem-solving into manageable parts.

Mastering Algebra Problem Solving

Algebra is a branch of mathematics that deals with symbols and rules for manipulating those symbols. When faced with a problem, the objective is to find the values that make the equations true. Here's how you can effectively approach algebra problem solving:

• Understand the problem: Read the exercise carefully and grasp the conditions given.
• Develop equations: Translate the word problem into mathematical expressions.
• Solve systematically: Use techniques like substitution or elimination to find the solution.

In our given problem, understanding translates to setting up equations based on total costs before and after price increases, and systematically solving them using substitution. Practice and persistence are key in mastering algebra problem-solving.

Conducting Price Increase Calculations

Price increase calculations are essential when you need to understand how changes affect costs. These calculations are commonly expressed in percentages.
Here's how to approach price increase calculations:

• Determine the increase rate: For example, orange juice increases by 25% and bagels by 20%.
• Calculate the new price: Add the percentage increase to the original amount. Or apply the multiplying factor directly: Original Price \(\times\) (1 + Increase rate).

In our problem, these calculations assist in forming the second equation representing the new total cost, showcasing how crucial price increase evaluation is in real-life budgeting and algebraic exercises.