Problem 44 - Algebra 1

Question

Your math teacher tells you that next week's test is worth 100 points and contains 38 problems. Each problem is worth either 5 points or 2 points. Because you are studying systems of linear equations, your teacher says that for extra credit you can figure out how many problems of each value are on the test. How many of each value are there?

Step-by-Step Solution

Answer

There are 8 problems of 5 points each and 30 problems of 2 points each on the test.

Step 1: Formulate the first equation

From the total number of problems, an equation can be formulated. Let's represent the 5-point problems as 'x' and the 2-point problems as 'y'. Since the total number of problems is 38, this can be represented as 'x + y = 38'.

Step 2: Formulate the second equation

We can derive a second equation from the total score of the test. Each 5-point problem is represented as 'x', each 2-point problem as 'y', and together, they form the total of 100 points. This leads to the second equation: '5x + 2y = 100'.

Step 3: Solve the system of equations

Two equations are now available: 'x + y = 38' and '5x + 2y = 100'. Solving this system of linear equations gives the values of x and y. By multiplying the first equation by 2, we get: '2x + 2y = 76'. Subtract this from the second equation to get '3x = 24', which leads to 'x = 8'. Substituting 'x = 8' in to the first equation gives 'y = 30'.

Key Concepts

Linear Equations

A linear equation is an algebraic expression that forms a straight line when graphed on a coordinate plane. It is characterized by constants and variables that are not raised to any power higher than one. In this exercise, linear equations help solve a practical problem involving test points.

Let's take a closer look at the two equations formulated from the problem:

• The first equation: \( x + y = 38 \). This equation represents the total number of problems, with \( x \) being the number of 5-point problems and \( y \) as the number of 2-point problems.
• The second equation: \( 5x + 2y = 100 \). This accounts for the total points, multiplying each problem type by their respective point value.

These equations are linear because they follow the form \( ax + by = c \), where \( a \), \( b \), and \( c \) are constants, and \( x \) and \( y \) are variables. They serve as the foundation for the systemic approach to problem solving.

Problem Solving

Problem solving using systems of linear equations provides a structured approach to finding unknown values through known relationships. Here, the system involves two equations describing constraints of the test problem. Understanding these constraints is fundamental.

Consider the process:

• Formulate Equations: Express the known data through equations that capture all given conditions (e.g., total number of problems and total points available).
• Solve the System: Use algebraic methods such as substitution or elimination to find the unknown variables. This involves manipulating one equation or combining both to isolate each variable.
• Validate Solutions: Once you calculate the values, check them against the original conditions to ensure they satisfy all constraints.

This systematic approach not only applies to test problems but extends to real-life scenarios where multiple variables influence outcomes.

Algebraic Expressions

Algebraic expressions are the language of mathematics, combining both numbers and variables through operations like addition, subtraction, multiplication, and division. They serve as representations of relationships and help express different scenarios mathematically.

In this exercise, algebraic expressions are used to represent:

• The total count of problems: \( x + y \).
• The total points accumulated: \( 5x + 2y \).

These algebraic expressions are vital in forming equations. By understanding how to manipulate these expressions, you can transform verbal problem descriptions into solvable mathematical sentences.

This involves:

• Identifying patterns in the problem to express relationships through variables.
• Simplifying or rearranging expressions to suit the problem-solving strategy.

Mastering algebraic expressions equips you with tools for tackling diverse mathematical challenges.