# The sum of two numbers is 180 and the larger number exceeds four times the smaller number by ten, what are the 2 numbers?

Oct 2, 2015

The numbers are $110$ and $70$.

#### Explanation:

Be $x$ and $y$ the two numbers. We know that

$x + y = 180$ and that

$x = y + 4 \cdot 10$

If we replace $x$ with $y + 40$ we find

$y + 40 + y = 2 y + 40 = 180$

$\rightarrow 2 y = 180 - 40 = 140$
$\rightarrow y = \frac{140}{2} = 70$

Then we find

$x = 70 + 40 = 110$

$\rightarrow x + y = 110 + 70 = 180$

Oct 2, 2015

The numbers are: $34$ and $146$

#### Explanation:

The problem is to find such numbers, that:

1. Their sum is 180
2. Larger number exceeds 4 times the smaller by 10.

These conditions lead to folowing system of equations:

$\left\{\begin{matrix}x + y = 180 \\ y = 4 x + 10\end{matrix}\right.$

If we substitute $y$ in yje first equation we get:

$x + 4 x + 10 = 180$
$5 x + 10 = 180$
$5 x = 170$
$x = 34$

Now we can substitute $x$ in any equation to calculate $y$:

$y = 4 \cdot 34 + 10$
$y = 136 + 10$
$y = 146$

Finally we can write the answer:

$\left\{\begin{matrix}x = 34 \\ y = 136\end{matrix}\right.$