# How do you solve the system 2x = y + 1 and 2x - y = 5?

Feb 28, 2016

You can either solve by elimination or by substitution. I'll solve by substitution

#### Explanation:

To perform substitution, you must first isolate one of the variables in one of the equations.

It would be easiest to isolate y in the first equation.

$2 x = y + 1 \implies 2 x - 1 = y$

Knowing y, we can substitute the value of y (2x - 1) for y in the other equation.

$2 x - \left(2 x - 1\right) = 5$

$2 x - 2 x + 1 = 5$

$0 x = 4$

$x = \frac{4}{0}$

$x = \emptyset$, since division by 0 is undefined in mathematics.

The solution set is $\left\{\emptyset\right\}$.

You could also have seen that there would be no solution by isolating y in both the original equations.

$2 x - 1 = y , 2 x - 5 = y$

As you can see, in both equations x is multiplied by 2, giving the same number as a result. However, in the first equation this value is subtracted by 1 while in the second it is subtracted by 5. This suggests two values of y for one value of x, which is impossible. So, at this point we can conclude that the lines are parallel and so they never intersect (in a systems of equations, the point of intersection of two lines is the solution).

If you have learned about linear functions, you should know that in slope intercept form, $y = m x + b$, the value of m is the same but the value of b is different when comparing the equations of lines that are parallel.

Hello from Esquimalt!

Hopefully this helps!