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

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

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Answer 1 · 2016-02-28T03:52:36

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.

$2 x = y + 1 \Rightarrow 2 x - 1 = y$ $2x = y + 1 => 2x - 1 = y$

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

$2 x - (2 x - 1) = 5$ $2x - (2x - 1) = 5$

$2 x - 2 x + 1 = 5$ $2x - 2x + 1 = 5$

$0 x = 4$ $0x = 4$

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

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

The solution set is ${\emptyset}$ ${O/}$ .

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

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

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, $y = m x + b$ $y = mx + b$ , the value of m is the same but the value of b is different when comparing the equations of lines that are parallel.

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

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