When you have a number of sets of linear equations like this, it can be convenient to make use of Cramer's Rule for finding the solution. It tells you the solution to
is given by
- x = (bf-ec)/(bd-ea)
- y = (cd-fa)/(bd-ea) . . . . . note the same denominator as for x
It is also convenient to put the equations in Standard Form, which has the leading coefficient positive and all numbers mutually prime. This form makes it easier to see dependent and inconsistent equation sets.
1. x +y = -1, 3x -y = -7
You can add these equations in your head to see 4x = -8; x = -2.
The solution is (-2, 1)
2. x -2y = -5; 5x +3y = 27
By Cramer's Rule, x = -39/-13 = 3; y = -52/-13 = 4.
The solution is (3, 4)
3. 4x -y = 9; 5x +2y = 3
By Cramer's Rule, x = 21/13 ≈ 1.615; y = -33/13 ≈ -2.538.
The solution is (21/13, -33/13) . . . . not among your offered choices
4. 2x +y = -2; 2x +y = -2
This is a dependent set, so has infinite solutions.
5. x -2y = 4; x -2y = -11/3
Note that we have fudged "standard form" a bit here in order to make the left sides of the equations the same. This points up the inconsistency of the equations.
This is an inconsistent set, so has no solution.
6. 7x -y = -1; 2x -y = 4
You can subtract the second equation from the first in your head to see 5x = -5; x = -1. Then y = 7x+1 = -6.
The solution is (-1, -6).