Finding Three Numbers Using Averages

This problem involves using the concept of averages to find the values of three unknown numbers. The average of a set of numbers is calculated by summing the numbers and dividing by the count of the numbers.

Let the three numbers be $x$, $y$, and $z$.

We are given the following information:

1. The average of the three numbers ($x, y, z$) is 6.
2. The average of the first two numbers ($x, y$) is 5.
3. The average of the last two numbers ($y, z$) is 8.

Let's translate these statements into mathematical equations:

• From statement 1: $\frac{x+y+z}{3} = 6$
• From statement 2: $\frac{x+y}{2} = 5$
• From statement 3: $\frac{y+z}{2} = 8$

Now, we can simplify these equations to find the sums of the numbers:

• Equation 1: $x+y+z = 6 \times 3 \implies x+y+z = 18$
• Equation 2: $x+y = 5 \times 2 \implies x+y = 10$
• Equation 3: $y+z = 8 \times 2 \implies y+z = 16$

We now have a system of three linear equations:

1. $x+y+z = 18$
2. $x+y = 10$
3. $y+z = 16$

We can solve this system to find the values of $x$, $y$, and $z$. A simple way is to use substitution.

Substitute the value of $x+y$ from Equation 2 into Equation 1:

$(x+y)+z = 18$

$10 + z = 18$

$z = 18 - 10$

$z = 8$

Now that we have the value of $z$, we can substitute it into Equation 3 to find the value of $y$:

$y+z = 16$

$y+8 = 16$

$y = 16 - 8$

$y = 8$

Finally, substitute the value of $y$ into Equation 2 to find the value of $x$:

$x+y = 10$

$x+8 = 10$

$x = 10 - 8$

$x = 2$

So, the three numbers are 2, 8, and 8.

Let's verify these numbers with the original conditions:

• Average of 2, 8, 8: $\frac{2+8+8}{3} = \frac{18}{3} = 6$. This matches the first condition.
• Average of the first two (2, 8): $\frac{2+8}{2} = \frac{10}{2} = 5$. This matches the second condition.
• Average of the last two (8, 8): $\frac{8+8}{2} = \frac{16}{2} = 8$. This matches the third condition.

All conditions are satisfied, so the three numbers are indeed 2, 8, and 8.

Revision Table: Key Concepts

Additional Information: Solving Systems of Equations

Solving a system of linear equations is a fundamental skill in algebra. There are several methods to solve them, including:

• Substitution Method: Solve one equation for one variable, and then substitute that expression into the other equations. This is the method used in the solution above. It's often useful when one equation can be easily solved for a variable.
• Elimination Method: Multiply equations by constants so that when the equations are added or subtracted, one variable cancels out. Repeat the process until one variable remains.
• Matrix Method: Represent the system of equations using matrices and use matrix operations (like finding the inverse matrix or using Gaussian elimination) to solve for the variables. This method is particularly useful for larger systems.

Choosing the best method depends on the specific system of equations you are trying to solve.