Question

The average of \(\\{-1,3,0,-2,4,2, x, y\\}\) is 3 and \(x-y=2 .\) What is the value of \(y ?\)

Short answer

The value of \(y\) is 8.

Step-by-step solution

Understand the average formula

The average of a set of numbers is calculated by dividing the sum of the numbers by the number of elements in the set. The formula for average is \[\text{Average} = \frac{\text{Sum of elements}}{\text{Number of elements}}.\]Given that the average is 3, there are 8 elements in the set.

Set up the equation for the sum

Since the average is 3 and there are 8 elements, the equation for the sum of the elements is\[Sum = 3 \times 8 = 24.\]

Create an equation using the set elements

The elements in the set are \{-1,3,0,-2,4,2,x,y\}. Their sum must be equal to 24. Therefore, we write the equation:\[-1 + 3 + 0 - 2 + 4 + 2 + x + y = 24.\]

Simplify the equation

Combine the known values on the left-hand side:\[6 + x + y = 24.\]Subtract 6 from both sides:\[x + y = 18.\]

Use the given relation between x and y

We know that \(x - y = 2\). We now have the system of equations:\[x + y = 18\]\[x - y = 2\]

Solve the system of equations

To solve the system, add the two equations:\[(x + y) + (x - y) = 18 + 2\]This simplifies to:\[2x = 20\]Divide both sides by 2:\[x = 10.\]

Find the value of y

Substitute \(x = 10\) back into the equation \(x + y = 18\):\[10 + y = 18.\]Subtract 10 from both sides to find \(y\):\[y = 8.\]

Key concepts

average calculation

The concept of finding the average is fundamental in understanding many math problems. The average of a set of numbers is calculated by summing all the numbers in the set and then dividing by the count of numbers.
For example, if we have the set of numbers \{-1,3,0,-2,4,2,x,y\}, and we are told the average is 3, we start by noting that there are 8 numbers in the set. The formula to find the average is given by: \[ \text{Average} = \frac{\text{Sum of elements}}{\text{Number of elements}}. \]
So in our case: \[3 = \frac{\text{Sum of elements}}{8}.\]
This equation helps us realize that: \[\text{Sum of elements} = 3 \times 8 = 24.\]
This is a crucial step that helps set up further calculations in the problem.

system of equations

A system of equations consists of multiple equations that need to be solved simultaneously. In this problem, we derive a system of two equations using the given information.
We know two key ideas: the sum of the elements is 24, and we have two unknown variables, x and y. From the given numbers in the set \{-1,3,0,-2,4,2,x,y\}, we set up the equation: \[-1 + 3 + 0 - 2 + 4 + 2 + x + y = 24\],
This simplifies to \[6 + x + y = 24\].
Then, subtract 6 from both sides, getting: \[x + y = 18\].
Further, we are given that: \[x - y = 2.\]
These two equations, \[x + y = 18\] and \[x - y = 2,\] create a system of linear equations that we need to solve together.

substitution method

Solving a system of equations can be done using different methods, one of which is the substitution method. This involves solving one of the equations for one variable and substituting that expression into the other equation.
In our problem, we already have the two equations: \[x + y = 18\] and \[x - y = 2.\]
We can solve them by adding both equations to eliminate y: \[(x + y) + (x - y) = 18 + 2,\]
which simplifies to: \[2x = 20.\]
Dividing both sides by 2 gives us: \[x = 10.\]
With x now known, substitute x back into one of the original equations, say \[x + y = 18,\] giving: \[10 + y = 18.\]
Subtraction yields: \[y = 8.\]

sum of elements

Understanding the sum of a set of elements is essential in various mathematical contexts. The sum of elements is simply the addition of all numbers present in a set.
Initially, we find from the average calculation that the sum of elements in our set is 24: \[\text{Sum of elements} = 3 \times 8 = 24.\]
Next, list the set elements: \{-1, 3, 0, -2, 4, 2, x, y\}.
Creating an equation, we get: \[-1 + 3 + 0 - 2 + 4 + 2 + x + y = 24.\]
Simplifying the constants: \[6 + x + y = 24.\]
Subtract 6 from both sides: \[x + y = 18.\]
This sum of the elements helps us eventually solve for the unknowns x and y, connecting various steps in the solution.