# The sum of two numbers is 4. Twice the larger is 11 more than the smaller. How do you find the smaller number?

Jan 5, 2017

The smaller number is $- 1$.

#### Explanation:

Considering the two numbers as $x$ and $y$ where $x$ is the larger number, we can write:

$x + y = 4$
$2 x = y + 11$

From the first equation, we can determine a value for $x$.

$x + y = 4$

Subtract $y$ from both sides.

$x = 4 - y$

In the second equation, substitute $x$ with $\textcolor{red}{\left(4 - y\right)}$.

$2 x = y + 11$

$2 \textcolor{red}{\left(4 - y\right)} = y + 11$

Open the brackets and simplify. The product of a positive and a negative is a negative.

$8 - 2 y = y + 11$

Add $2 y$ to both sides.

$8 = 3 y + 11$

Subtract $11$ from both sides.

$- 3 = 3 y$

Divide both sides by $3$.

$- 1 = y$ or $y = - 1$

In the first equation, substitute $y$ with $\textcolor{b l u e}{- 1}$.

$x + y = 4$

$x + \left(\textcolor{b l u e}{- 1}\right) = 4$

Open the brackets and simplify. The product of a positive and a negative is a negative.

$x - 1 = 4$

Add $1$ to both sides.

$x = 5$