Fiona bought some socks that cost $4.95 for each pair and some belts that cost $6.55 each. Fiona spent $27.95 in all. Let A represent the number of pairs of socks purchased.

Fiona bought some socks that cost $4.95 for each pair and some belts that cost $6.55 each. Fiona spent $27.95 in all. Let A represent the number of pairs of socks purchased and B the number of belts purchased.

We can write this as 4.95A + 6.55B = 27.95

We can model this situation with a system of equations. Let:

• A represent the number of pairs of socks purchased
• B represent the number of belts purchased

The total cost of the socks is 4.95A, and the total cost of the belts is 6.55B.

We know that the total cost of both items is $27.95, so we can write the following equation:

4.95A + 6.55B = 27.95

This system of equations can be used to solve for the unknown values of A and B.

System of Linear Equations in Algebra

A system of linear equations is a collection of two or more linear equations that involve the same variables. Each linear equation represents a straight line, and the system is solved to find the values of the variables that satisfy all the equations simultaneously.

Here are some key points about systems of linear equations:

• Variables: These are the unknowns that we are trying to solve for. They are represented by letters like x, y, z, etc.
• Coefficients: These are the numerical factors that multiply the variables in each equation.
• Constant terms: These are the numerical values that are added or subtracted to the variables on one side of the equation.
• Solution: A solution is a set of values for the variables that makes all the equations in the system true when substituted in.
• Types of solutions: A system can have:
  • Unique solution: This is when there is only one set of values that satisfies all the equations. This happens when the lines representing the equations intersect at a single point.
  • Infinitely many solutions: This occurs when the lines representing the equations overlap completely. Every point on the overlapping line is a solution.
  • No solution: This happens when the lines representing the equations are parallel but not overlapping. No point satisfies both equations simultaneously.

There are various methods to solve systems of linear equations, including:

• Elimination method: This involves manipulating the equations to eliminate one of the variables and then solving for the remaining variable(s).
• Substitution method: This involves solving one equation for one variable and substituting that expression into another equation to solve for the remaining variable(s).
• Graphical method: This involves plotting the lines represented by each equation and finding the point(s) where they intersect.

Understanding systems of linear equations is a fundamental concept in algebra and has applications in various fields, including physics, chemistry, economics, and computer science.