Question

Your teacher is giving a test worth 250 points. There are 68 questions. Some questions are worth 5 points and the rest are worth 2 points. How many of each question are on the test?

Short answer

There are 42 questions worth 5 points and 26 questions worth 2 points on the test.

Step-by-step solution

Define Variables

Let's call the number of 5-points questions as \(x\) and the number of 2-points questions as \(y\).

Create Equations

From the problem, we can create two equations: \n1. \(x+y=68\) (since the number of both types of questions sums to 68) \n2. \(5x + 2y = 250\) (since the worth of all questions sum to 250)

Solve the System

Substitute the value of \(y\) from equation 1 into equation 2 and solve for \(x\), \nSubstitute \(y = 68 - x\) in the second equation \nThen we have, \(5x + 2*(68 - x) = 250\), from which we find that \(x = 42. \nThen substitute \(x = 42\) back into equation 1 to find \(y = 68 - 42 = 26\).\nSo, there are 42 questions worth 5 points and 26 questions worth 2 points.

Key concepts

Algebraic Problem Solving

When facing an algebraic problem, the first step is to identify what you're trying to find. In our example, you need to figure out how many questions of each point value are on a test. Once identified, assign variables to represent these unknowns. Typically, variables like 'x' and 'y' are used as placeholders for the quantities we are trying to determine.

After setting up variables, translate the words of the problem into mathematical equations that represent the relationships between the quantities involved. In the given exercise, two key relationships were identified: the total number of questions and the total point value of all questions. These form the basis of our equations.

To solve the problem, use a combination of algebraic techniques, such as simplifying expressions, combining like terms, and performing operations that will help isolate the variables. By systematically applying these techniques and logical reasoning, you arrive at the solution, thus making algebraic problem solving a fundamental skill in mathematics, especially for standardized testing and real-world applications.

Linear Equations

Linear equations are an essential component of algebra that represent a straight line when graphed on a coordinate plane. These equations can usually be written in the form \(Ax + By = C\), where 'A', 'B', and 'C' are constants and 'x' and 'y' are variables. In our textbook problem, the two equations \(x + y = 68\) and \(5x + 2y = 250\) are both linear, as they can be rearranged to this standard form.

Characteristics of Linear Equations

Each linear equation represents a relationship where one variable is a constant multiple of another, or both are related in a way that their collective outcomes are linear. They show directly proportional or inversely proportional relationships depending on their coefficients.

Understanding the properties of linear equations is critical when solving a system involving multiple equations, as it helps determine the most efficient method, such as substitution or elimination, to find the solution set.

Variable Substitution

Variable substitution is a powerful tool in algebra, especially when dealing with systems of equations. It involves replacing one variable with another expression obtained from a different equation, thereby reducing the number of variables and simplifying the system.

In our test example, we used substitution to solve the system. Once we expressed \(y\) in terms of \(x\) from the first equation \(y = 68 - x\), we substituted this expression for \(y\) in the second equation. This action transformed the equation into one variable, making it solvable through basic algebraic maneuvers.

Advantages of Variable Substitution

• Simplifies complex systems to single-variable equations
• Facilitates the use of algebraic operations to isolate and solve for the remaining variable
• Integrates the information from multiple equations

By mastering variable substitution, students can tackle a wide range of problems, from basic algebra to more advanced topics involving complex systems.