Problem 51 - Introductory Algebra for College Students

Question

The average, or mean, \(A,\) of three exam grades, \(x, y,\) and \(z,\) is given by the formula $$A=\frac{x+y+z}{3}$$ a. Solve the formula for \(z\). b. Use the formula in part (a) to solve this problem. On your first two exams, your grades are \(86 \%\) and \(88 \%\) : \(x=86\) and \(y=88 .\) What must you get on the third exam to have an average of \(90 \% ?\)

Step-by-Step Solution

Answer

The score you need on the third exam to have an average of 90% is 96%.

Step 1: Solve the formula for \(z\)

Start from the formula given for the average, \(A = \frac{x + y + z}{3}\). Multiplying both sides by 3, we get \(3A = x + y + z\). Then solving for \(z\) gives \(z = 3A - x - y\).

Step 2: Substitute the given values

Now we substitute our given values \(A=90\), \(x=86\), \(y=88\) into the formula we got for \(z\). We get \(z = 3*90 - 86 - 88\).

Step 3: Solve for \(z\)

We'll then solve the equation to find the value of \(z\), which is \(z = 270 - 86 - 88 = 96\).

Key Concepts

Solving Algebraic Equations

Solving algebraic equations is a foundational skill in algebra. It involves finding the value(s) of variable(s) that make the equation true. One common approach is to perform operations that move terms from one side of the equation to the other, aiming to isolate the variable.

Let's consider a practical example: finding the grade needed on the third exam to achieve an overall average. We start with the mean formula, \(A = \frac{x + y + z}{3}\), and manipulate it to solve for \(z\). Multiplying both sides by 3 to eliminate the denominator, we get \(3A = x + y + z\). The final step to isolate \(z\) involves subtracting \(x\) and \(y\) from both sides, resulting in \(z = 3A - x - y\). This process of isolating variables is at the heart of solving algebraic equations.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can contain numbers, variables, and operations. They represent quantities without an equality sign, unlike equations. For instance, in our problem, \(3A - x - y\) is an expression that represents the score needed on the third exam to achieve a mean of \(A\).

Understanding and manipulating these expressions are crucial. It allows you to reconfigure the initial mean formula into a new form that can give you specific insights - such as the score needed on a future exam.

Mathematical Problem-Solving

Mathematical problem-solving involves a series of steps that help you understand and solve mathematical questions. These include understanding the problem, devising a plan, carrying out the plan, and checking your solution.

Our example showcases this process: First, we understand that the problem requires us to find the third exam score to achieve a particular average. The plan involves modifying the average formula to solve for the unknown score. After performing the algebraic steps, we reach a solution and we can check it by plugging \(z\) back into the initial formula to verify that it yields the desired average.

College Algebra

College algebra is an advanced course that builds upon high school algebra. It involves more complex algebraic concepts and problem-solving techniques. In this context, the problem of finding the required third exam score involves skills that are often honed in a college algebra course.

Students learn about variables, algebraic expressions, and how to solve equations—everything necessary to tackle real-world problems, like calculating exam grades to determine what is needed to reach a goal average. This demonstrates the practical application of college algebra in everyday situations.