Problem 88 - College Algebra Essentials

Question

Suppose that we agree to pay you 8e for every problem in this chapter that you solve correctly and fine you \(5 \phi\) for every problem done incorrectly. If at the end of 26 problems we do not owe each other any money, how many problems did you solve correctly?

Step-by-Step Solution

Answer

The number of problems correctly solved is the value of x obtained in step 2.

Step 1: Set up the equations

Let \(x\) represent the problems solved correctly and \(y\) represent the problems solved incorrectly. The total number of problems \(x + y = 26\) (Equation 1). For the payments, you gain 8e for each correct problem and lose \(5 \phi\) for each incorrect problem, and the total is zero. Therefore, the equation is \(8ex-5 \phi y = 0\) (Equation 2).

Step 2: Solve the system

First, express \(y\) from Equation 1 as a function of \(x\), so \(y = 26-x\), then substitute this into Equation 2: \(8ex-5 \phi(26-x) = 0\). Solve the equation to find the value of \(x\).

Step 3: Find the value of y

Use the value of \(x\) got from step 2 to find the value of \(y\) by substituting \(x\) into our first equation \(y = 26-x\). This will give you the number of problems incorrectly solved.

Step 4: Verify the solution

To make sure the solution is correct; the value of \(x\) and \(y\) should satisfy both Equation 1 and Equation 2. If these values satisfy both equations, it means the solution is correct.

Key Concepts

Algebraic Problem-Solving

Algebraic problem-solving is like a toolkit for cracking numerical puzzles. It involves manipulating equations and expressions using algebraic rules to find unknown values. In our exercise, the goal was to determine the correct number of problems solved given a certain financial reward and penalty structure.

First, identifying variables is critical. We let 'x' represent the number of problems solved correctly, and 'y' as the problems solved incorrectly. With this setup, translating the conditions of the problem into mathematical language was the next step, leading to two key equations. Mastering algebraic problem-solving includes setting up equations, substituting values, and systematically working towards the solution.

Improving problem-solving skills in algebra often means practicing setting up equations accurately, logically thinking through the relationships between quantities, and consistently checking for errors by substituting found solutions back into original equations to verify their correctness. It's also helpful to break down complex problems into smaller, more manageable parts, just as we separate finding 'x' from finding 'y' in this exercise.

Linear Equations

Linear equations form the backbone of simultaneous equations, and they're described as equations of the first order. These are neat, straight-line relationships between variables with no exponents or powers involved. In the form of 'ax + by = c', 'a' and 'b' are coefficients, while 'c' is the constant.

The beauty of linear equations lies in their predictability and the ease with which we can graph them. In the context of our question, '8ex - 5\(\phi\)y = 0' and 'x + y = 26' are two linear equations that map out a clear relationship between 'x' and 'y'. The process of solving them involves isolating one variable and substituting its value into the other equation, which is the essence of algebraic manipulation.

To excel in working with linear equations, one must become fluent in the methods of solving for variables, like using substitution or the addition method. This skill is fundamental as linear equations are ubiquitous in mathematics and its applications.

Simultaneous Equations

Finding the Intersection

Solving simultaneous equations means finding the point where two or more equations agree - think of it as the shared solution where the lines intersect on a graph. In our exercise, the equations 'x + y = 26' and '8ex - 5\(\phi\)y = 0' need to be satisfied at the same time, which meant finding the values of 'x' and 'y' that make both equations true.

There are multiple strategies for solving simultaneous equations, such as substitution, elimination, and graphical methods. Substitution involves expressing one variable in terms of another and then substituting it into the other equation, while elimination aims to add or subtract equations to cancel out one of the variables.

Practice Makes Perfect

Understanding simultaneous equations develops critical thinking and the ability to visualize abstract concepts. The key to mastering them is practice, which helps in recognizing patterns and developing strategies to efficiently solve the equations. Remember, the answers must satisfy all the given equations - this verification step is crucial to ensure the correctness of the solution.