The sum of two numbers is 4 less than twice the larger. If the larger is decreased by three times the smaller, the result is -20. Find the numbers.

Steps to Solve

1. Define Variables

Let the larger number be $x$ and the smaller number be $y$.

2. Set Up the First Equation

According to the problem, the sum of the two numbers is 4 less than twice the larger number. This gives us the equation:

$$ x + y = 2x - 4 $$

3. Rearrange the First Equation

Let's rearrange the first equation to isolate $y$:

$$ y = 2x - 4 - x $$

Simplifying, we get:

$$ y = x - 4 $$

4. Set Up the Second Equation

The second condition states that if the larger number is decreased by three times the smaller number, the result is -20. This leads to the equation:

$$ x - 3y = -20 $$

5. Substitute for y

Now, substitute the expression for $y$ from the first equation into the second equation:

$$ x - 3(x - 4) = -20 $$

6. Simplify and Solve for x

Distributing the $-3$ gives us:

$$ x - 3x + 12 = -20 $$

Combine like terms:

$$ -2x + 12 = -20 $$

Subtract 12 from both sides:

$$ -2x = -32 $$

Now divide by -2:

$$ x = 16 $$

7. Find y

Now that we have $x$, substitute it back into the equation for $y$:

$$ y = 16 - 4 = 12 $$

8. Final Values

The two numbers are $x = 16$ and $y = 12$.