Question

The sums of any two of three real numbers are 24, 28, and 30. Find these three numbers.

Short answer

The three numbers are $\mathrm{x}=13,\mathrm{y}=11$and $\mathrm{z}=17$.

Step-by-step solution

Making the equation

We have three real unknown numbers, say x, y, and z.

We have sums of any pair of those that are 24, 28, and 30 and the pairs are (x, y), (y, z), and (x, z)

We construct the system;

$$\begin{gathered} \mathrm{x}+\mathrm{y}=24 \\ \mathrm{y}+\mathrm{z}=28 \\ \mathrm{x}+\mathrm{z}=30 \end{gathered}$$

Eliminating the variables

Now we use the R₃ ->R₃-R₁

From here 2z = 34

z = 17

From the second row, we get

$\begin{array}{rcl}\mathrm{y}+\mathrm{z}& =& 28\\ \mathrm{y}+17& =& 28\\ \mathrm{y}& =& 11\end{array}$

Then the first row

$\begin{array}{rcl}\mathrm{x}+\mathrm{y}& =& 24\\ \mathrm{x}& =& 11=24\\ \mathrm{x}& =& 13\end{array}$

Final answer

Thus, the required numbers are $\mathrm{x}=13,\mathrm{y}=11$and z = 17.