Hugh bought some magazines that cost $3.95 each and some books that cost $8.95 each. He spent a total of $47.65. Let m represent the number of magazines, b represent the number of books.

Hugh bought some magazines that cost $3.95 each and some books that cost $8.95 each. He spent a total of $47.65. Let m represent the number of magazines and b represent the number of books. Which equation models the situation?

The equation that models the situation is 3.95m + 8.95b = 47.65.

Let m represent the number of magazines and b represent the number of books.

The cost of m magazines at $3.95 each is 3.95m. The cost of b books at $8.95 each is 8.95b.

Since the total amount spent is $47.65, we can write the equation:

3.95m + 8.95b = 47.65

So, the equation that models the situation is 3.95m + 8.95b = 47.65.

System of Linear Equations in Algebra

A system of linear equations in algebra, also called a linear system, refers to a collection of two or more linear equations that involve the same variables.

Here's a breakdown of the key points:

• Linear equation: An equation where the highest power of each variable is 1. It can be written in the general form of ax + by = c where a, b, and c are constants and x and y are the variables.
• Variable: An unknown quantity represented by a letter, like x, y, or z.
• Solution: A set of values for the variables that makes all the equations in the system true simultaneously.

The following are some important aspects of systems of linear equations:

• Number of solutions: Depending on the relationships between the equations, a system can have:
  • One unique solution: This occurs when the lines representing the equations intersect at a single point.
  • Infinitely many solutions: This happens when the lines representing the equations coincide (completely overlap).
  • No solution: This occurs when the lines representing the equations are parallel but not coincident.
• Solving methods: There are various methods for solving systems of linear equations, including:
  • Substitution method: Solve one equation for a variable and substitute that expression into the other equation.
  • Elimination method: Manipulate the equations to eliminate one variable and then solve for the remaining variable.
  • Matrix method: Represent the system as a matrix and perform operations to solve for the variables.

Understanding systems of linear equations forms the foundation of linear algebra, a branch of mathematics with applications in various fields, including physics, engineering, economics, and computer science.