Answer:
(b) There are infinitely many solutions
Explanation:
You want to know the number of solutions to the system of equations ...
Dependent
A set of equations is dependent, and has infinitely many solutions, if one of the equations can be obtained from some combination of the others.
Here, dividing the first equation by 2 gives you ...
8/2x -2/2y =-4/2
4x -y = -2
This equation is identical to the second equation, so every one of the infinitely many solutions to this will also be a solution to that.
The system has infinitely many solutions.
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Additional comment
When the equations are written in standard form (or general form), the determinant of the variable coefficients can tell you a story. For equations ...
The determinant of the coefficients is ae -db.
For your equations, this is 8(-1) -(4(-2)) = -8 +8 = 0.
When the determinant is zero, there will not be a unique solution. There will be either none or infinitely many.
You can perform the same calculation on another two columns of the coefficients to see which:
bg -ec = -2(-2) -(-1)(-4) = 4 -4 = 0
When this value is also zero, it means the system is dependent and has infinitely many solutions. (If it is non-zero, the system is inconsistent and has no solutions.)
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