# What is the answer of 4y 2y

## Equation Method - Finding the amount of solution by equating the equations

We also do that with the equation procedure. We rearrange both equations for a variable, for example y, and because y should be the same for both, we can set the equations equal.

We do that with an example (where we get a solution):

2x + y = 1 | - 2x

y = - 2x + 1

- x + y = - 2 | + x

y = x - 2

Now we set both equations equal and can then solve for x:

- 2x + 1 = x - 2 | + 2x

1 = 3x - 2 | + 2

3 = 3x | : 3

1 = x

Now we can insert our calculated x into one of the two original equations and calculate y, i.e. either in y = - 2x + 1 or in y = x - 2 or in both, because the same y must come out in both equations so that one that can count as a sample. We get the solution set: L = {(1 | - 1)}

We want to calculate another example, this time we will not get a solution, i.e. the empty set as the solution set. The example:

2x + 4y = 4

x + 2y = 6

We calculate according to the scheme: First of all, convert both equations to y:

2x + 4y = 4 | - 2x

4y = 4 - 2x | : 4

y = 1 - 0.5x

x + 2y = 6 | - x

2y = 6 - x | : 2

y = 3 - 0.5x

Equate:

1 - 0.5x = 3 - 0.5x | + 0.5x

1 = 3

We get a contradiction here, because 1 is not equal to 3. If we get a contradiction, then our solution set is equal to the empty set. Or written in formulas: L = Ø

At the end of the equation procedure, we also want to calculate an example with an infinite number of solutions, the example:

2x - 4y = - 12

- 3x + 6y = 18

We switch the equations to y again one after the other (we could switch to x too, but we do it with y).

2x - 4y = - 12 | - 2x

- 4y = - 12 - 2x | : (- 4)

y = 3 + 0.5x

- 3x + 6y = 18 | + 3x

6y = 18 + 3x | : 6

y = 3 + 0.5x

We can already see that we get the same straight line twice, but we still set the same:

3 + 0.5x = 3 + 0.5x | - 0.5x

3 = 3

If at the end of the equation we get a true statement that is independent of the variable, then we have the case that there are infinitely many solutions. The solution set is the straight line: y = 3 + 0.5x.

We have now switched to y every time and equated, we want to see that we get the same result when we switch to x:

2x - 4y = - 12 | + 4y

2x = - 12 + 4y | : 2

x = - 6 + 2y

- 3x + 6y = 18 | - 6y

- 3x = 18 - 6y | : (- 3)

x = - 6 + 2y

Now we set the same again:

- 6 + 2y = - 6 + 2y | - 2y

– 6 = – 6

We get a true statement again regardless of the variable. So there are infinitely many solutions and the solution set is a straight line, namely x = - 6 + 2y.

By the way, this is the same result as before, we have to be careful that we now substitute for y and get x - not as usual. But let's compare the straight lines:

x = - 6 + 2y and

y = 3 + 0.5x

They should be the same, so we solve the first straight line for y:

x = - 6 + 2y | + 6

x + 6 = 2y | : 2

0.5x + 3 = y

And we see that it's true.

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