The cost of ten apples, eight kiwis and 12 papaya is ₹240. The cost of eight apples, six kiwis and ten papayas is ₹ 180. Find the cost of one apple, one kiwi and one papaya. 

The cost of ten apples, eight kiwis and 12 papaya is ₹240. The cost of eight apples, six kiwis and ten papayas is ₹ 180. Find the cost of one apple, one kiwi and one papaya.

Let's denote the cost of an apple, a kiwi, and a papaya as A, K, and P respectively.

We are given a system of two equations with three unknowns:

1. 10A + 8K + 12P = 240 (Equation 1)
2. 8A + 6K + 10P = 180 (Equation 2)

To solve for the cost of one apple, one kiwi, and one papaya (A + K + P), we need to eliminate one of the variables. Since we have two equations and three unknowns, eliminating one variable is possible.

Here, we can eliminate P by subtracting Equation 2 from Equation 1:

(1) 10A + 8K + 12P = 240

(-) (2) 8A + 6K + 10P = 180

2A + 2K + 2P = 60

Dividing both sides by 2, we get:

A + K + P = 30

Therefore, the cost of one apple, one kiwi, and one papaya is ₹30.

Systems of Linear Equations

A system of linear equations is a collection of two or more linear equations, which involve the same variables. A linear equation is an equation where the highest power of each variable is 1.

Here are some key points about systems of linear equations:

• Variables: Each equation in the system involves the same set of variables. These variables represent unknown quantities that we want to solve for.
• Solution: A solution of a system of linear equations is a set of values for the variables that satisfies all the equations in the system simultaneously. In other words, if you plug these values into each equation, you get a true statement.
• Number of solutions: Depending on the relationships between the equations, a system can have:
  • One unique solution: This is the most common case, where the lines (for 2 variables) or planes (for 3 variables) representing the equations intersect at a single point.
  • Infinitely many solutions: This occurs when the equations represent lines (or planes) that completely overlap, indicating any point on that line (or plane) satisfies both equations.
  • No solution: This happens when the lines (or planes) are parallel, indicating they never intersect.

Example:

Here's an example of a system of linear equations in two variables:

• Equation 1: 2x + 3y = 5
• Equation 2: x - y = 1

This system can be solved using various methods like:

• Graphical method: Plotting both equations and finding the point of intersection.
• Substitution method: Solving one equation for one variable and substituting it into the other equation to solve for the remaining variable.
• Elimination method: Manipulating the equations to eliminate one variable and then solving for the remaining variable.

These methods will help you find the solution, which in this case is:

• x = 2
• y = 1