# One number is greater than another by fifteen, if 5 times the larger number minus twice the smaller one is three? find the two numbers.

Apr 19, 2018

(-9,-24)

#### Explanation:

First set up a system of equations:

Set the bigger number to x and the smaller number to y

Here are the two equations:

$x = y + 15$ Note that you add 15 to y because it is 15 smaller than x.

and

$5 x - 2 y = 3$

From here there are a few ways to solve this system. The fastest way however would be to multiply the entire first equation by -2 to get:

$- 2 x = - 2 y - 30$

rearranging this gives

$- 2 x + 2 y = - 30$

$- 2 x + 2 y = - 30$ and
$5 x - 2 y = 3$

You can now simply add the two functions together and cancel out the y term. This gives a single variable equation that you can solve:

$3 x = - 27$
Solving this gives x=-9

With your x value, you can now plug it in to either equation (whichever one you find easiest to work with) and solve for y.

$\left(- 9\right) = y + 15$ Insert x=-9
$- 9 = y + 15$ Subtract 15 from both sides
$- 24 = y$

Now you have both your x and y values that satisfy the equation. You write them in the coordinate pair (-9,-24)

To check your answer, go back and plug in both numbers to the other equation.

$5 \left(- 9\right) - 2 \left(- 24\right) = 3$
$- 45 + 48 = 3$
$3 = 3$