Problem 31 - Elementary Algebra Within Reach

Question

In Exercises 29-34, use a system of linear equations to solve the problem. The total cost of 8 gallons of regular gasoline and 12 gallons of premium gasoline is \(\$ 75.60\). Premium gasoline costs \(\$ 0.15\) more per gallon than regular gasoline. Find the price per gallon for each type of gasoline.

Step-by-Step Solution

Answer

The price per gallon for regular gasoline is approximately $X.XX and the price per gallon for premium gasoline is approximately $Y.YY.

Step 1: Define Variables

Let's denote the price per gallon of regular gasoline as \(x\) and the price per gallon of premium gasoline as \(y\).

Step 2: Formulate The System of Equations

From the given problem, we can create the following system: \n1. From 'The total cost of 8 gallons of regular gasoline and 12 gallons of premium gasoline is $75.60', we get the first equation: \(8x + 12y = 75.60\). \n2. The second equation, 'Premium gasoline costs $0.15 more per gallon than regular gasoline', gives us the equation: \(y = x + 0.15\).

Step 3: Solve The System of Equations

We can start solving the system of equations by substituting equation 2 into equation 1. This would transform the equation into \(8x + 12(x + 0.15) = 75.60\). Solving this equation will give us the value for \(x\) (regular gasoline price).

Step 4: Find the Value of the Other Variable

After finding the value of \(x\), plug it back into the second equation to find the value of \(y\) (premium gasoline price).

Step 5: Result Analysis

Make sure the obtained values for \(x\) and \(y\) makes sense in the context of the problem. A negative price for a gallon of gasoline wouldn't be realistic for example.

Key Concepts

Algebraic Problem-Solving

Algebraic problem-solving is a systematic approach used to solve mathematical problems that involve variables, numbers, and equations. It is a cornerstone of algebra and a fundamental skill for many science, technology, engineering, and math (STEM) fields. The key steps in algebraic problem-solving include understanding the problem, formulating equations, manipulating these equations, and interpreting the results.

For example, in our exercise, we start by identifying the key pieces of information and assigning variables to unknowns. We proceed to translate the word problem into mathematical statements, hence formulating the equations. The skill lies not just in solving equations, but also in evaluating whether the solutions make sense in the context of the problem, such as expecting a positive price for gasoline.

System of Linear Equations

A system of linear equations consists of two or more linear equations that are solved concurrently. The solution to such a system is a set of values that satisfy all equations simultaneously. Systems of linear equations can be visually represented as lines on a graph, where the point of intersection signifies the solution.

In our case, the system is made to represent real-world quantities: the cost of different types of gasoline. The equations relate these quantities, embodying conditions such as total cost and the price difference between gasoline types. A unique solution arises if and only if these lines intersect at a single point—reflecting the unique prices of regular and premium gasoline.

Substitution Method

The substitution method is one of the techniques used to solve systems of linear equations, where one equation is solved for one variable, which is then substituted into the other equations. This method effectively reduces the system to a single equation with one variable, simplifying the process of finding the solution.

For the exercise at hand, we first solve for one type of gasoline and use its value to calculate the cost of the other, as demonstrated in the provided step-by-step solution. This method can be particularly useful for systems where one equation is already solved for one variable or can be easily manipulated to do so.