Question

One number is 9 times more then 3 times another. their product is 9 more than 3 times their sum.

Answer 1

Let's call one of the numbers as x and the other as y.

From the first sentence

"One number is 9 more than 3 times another."

From this sentence, we know that our first number(x) plus 9 is equal to 3 times the other(y).

`x+9=3y`

From the other sentence

"Their product is 9 more than 3 times their sum"

We get the following equation

`xy+9=3(x+y)`

Now we have two equations for two variables

`\begin{cases}x+9=3y \\ xy+9=3(x+y)\end{cases}`

Expanding the parentheses, we have

`\begin{cases}x+9=3y \\ xy+9=3x+3y\end{cases}`

Using our value for 3y in the first equation in the second equation, we get a new equation

`\begin{gathered} xy+9=3x+3y \\ xy+9=3x+(x+9) \\ xy+9=3x+x+9 \\ xy+9=4x+9 \\ xy=4x \end{gathered}`

This last equation have an immediate solution, x = 0. If x is not 0, we can divide both sides by x.

`(xy)/(x)=(4x)/(x)\Rightarrow y=4`

Using this value for y, we can evaluate any of the equations to find its corresponding x, and we can do the same for x = 0 to find the corresponding value for y.

`\begin{gathered} y=4\Rightarrow x+9=3\cdot4\Rightarrow x=3 \\ x=0\Rightarrow0+9=3y\Rightarrow y=3 \end{gathered}`

This means, our points are (0, 3) and (3, 4).