-4x+y+=1

2y-2=8x

Is there a typo in the first equation?

I guess you mean

y - 4 x = 1
and
y - 4 x = 1
Those two lines are quite similar, in fact they intersect everywhere and there is no single point where they intersect.
This is a trick problem and there is no solution.

or to say it better there is an infinite number of solutions. For any real x there is a real y on the line.

To solve this system of linear equations, you can use either the substitution method or the elimination method. Let's use the elimination method:

Step 1: Rearrange the equations to have the same coefficients for either the x or y term.
Equation 1: -4x + y = 1 --> Multiply by 2 to get -8x + 2y = 2
Equation 2: 2y - 2 = 8x

Step 2: Now that the coefficients of the x term are the same, you can eliminate x by adding both equations together.
(-8x + 2y) + (8x) = 2 + (8x)
The x terms will cancel out, giving you: 2y = 8x + 2.

Step 3: Simplify the equation.
2y = 8x + 2
Divide by 2 to isolate y: y = 4x + 1.

Step 4: Substitute the value of y into one of the original equations to solve for x.
Let's use Equation 1: -4x + y = 1.
Substitute y with 4x + 1: -4x + (4x + 1) = 1.
Simplify: -4x + 4x + 1 = 1.
The x terms will cancel out, giving you: 1 = 1.

Step 5: Check if the equations are consistent or inconsistent.
In this case, both equations are consistent since the result is a true statement (1 = 1). This means the system of equations has infinitely many solutions.

To summarize, the system of equations has infinitely many solutions. The solution can be represented as y = 4x + 1.