How do you solve 4x + y = 4 and y = 4 - 4x using substitution?

May 12, 2017

Both $x$ and $y$ equal all real numbers

Explanation:

If $y = 4 - 4 x$, then we can replace the $\textcolor{b l u e}{y}$ in $4 x + \textcolor{b l u e}{y} = 4$ with $4 - 4 x$:

$4 x + \left(4 - 4 x\right) = 4$

Solve for $x$

$4 x + 4 - 4 x = 4$

subtract $4$ on both sides

$4 x - 4 x = 0$

$0 x = 0$

$x = \infty$
$\textcolor{w h i t e}{0}$

Let's try solving for $y$ now.

We need to isolate $x$:

$4 x + y = 4$

$4 x = 4 - y$

$x = \frac{4 - y}{4}$

Now, let's replace $\textcolor{b l u e}{x}$ with $\frac{4 - y}{4}$ in $y = 4 - 4 \textcolor{b l u e}{x}$

$y = 4 - \cancel{4} \left(\textcolor{b l a c k}{\frac{4 - y}{\cancel{4}}}\right)$

$y = 4 - 4 - y$

$0 y = 0$

$y = \infty$

The reason $x$ and $y$ both equal $\infty$ is that any number, multiplied by $0$, equals $0$. So, $x$ can equal $- 500$, $\frac{8}{19}$, or $9.01 \times {10}^{13}$, and still equal $0$, thus making the equation true. The same fact also applies for $y$.

Both $x$ and $y$ equal all real numbers