# If four times a first number is decreased by three times the second number, the result is zero. The sum of the numbers is -7. Find the numbers?

Apr 16, 2017

$- 3 , \mathmr{and} , - 4.$

#### Explanation:

By what is given,

$4 {N}_{1} - 3 {N}_{2} = 0. . . \left(1\right) , \mathmr{and} , {N}_{1} + {N}_{2} = - 7. . . \left(2\right) .$

$\left(2\right) \Rightarrow 3 \left({N}_{1} + {N}_{2}\right) = - 21 , i . e . , 3 {N}_{1} + 3 {N}_{2} = - 21. . . \left(2 '\right) .$

$\left(1\right) + \left(2 '\right) \Rightarrow 4 {N}_{1} + 3 {N}_{1} = 0 + \left(- 21\right) ,$

$\Rightarrow 7 {N}_{1} = - 21 \Rightarrow {N}_{1} = - \frac{21}{7} = - 3 , \text{ &, by (2), then, } {N}_{2} = - 4.$

Apr 16, 2017

$- 3$ and $- 4$

#### Explanation:

Let's say the first number is $x$ and the second number is $y$

So, first we're told that $4 \cdot x - 3 \cdot y = 0$ and then, that $x + y = - 7$

Let's use substitiution!

Our first job is to set $x + y = - 7$ to $x$:
$x + y = - 7$
subtract $y$ on both sides
$x = - 7 - y$

Now, substitute $x$ for $\left(- 7 - y\right)$ in the firstequation:
$4 x - 3 y = 0$
$4 \left(- 7 - y\right) - 3 y = 0$
distribute the $4$
$- 28 - 4 y - 3 y = 0$
add $28$ on both sides
$- 4 y - 3 y = 28$
combine like-terms
$- 7 y = 28$.
divide by $- 7$ on both sides
$y = - 4$

Now that we know $y = - 4$, we can solve for $x$:
$x + y = - 7$
$x + \left(- 4\right) = - 7$
add $4$ on both sides
$x = - 3$

Just to double-check our work, let's solve the first equation using $- 3$ and $- 4$ as $x$and $y$
$4 x - 3 y = 0$

If our answers are correct, we should find our solution to be $0$!
$4 \left(- 3\right) - 3 \left(- 4\right)$
$- 7 - - 7$
$- 7 + 7$
$0 = 0$
We were right!