4. If x = y + 3, then in the second equation you can write
-3 (y + 3) + 3y = 6
Expanding and simplifying gives
-3y - 9 + 3y = 6
-9 = 6
but this is clearly not true, so this system as no solution.
5. Solve the first equation for x :
x + 2y = 4 ===> x = -2y + 4
Substitute into the other equation:
2 (-2y + 4) - 3y = 1
-4y + 8 - 3y = 1
-7y = -7
y = 1
Solve for x :
x = -2•1 + 4
x = -2 + 4
x = 2
6. Solve the first equation for y :
2x - y = 6 ===> y = 2x - 6
Substitute into the other equation:
x + 2 (2x - 6) = -2
x + 4x - 12 = -2
5x = 10
x = 2
Solve for y :
y = 2•2 - 6
y = 4 - 6
y = -2
7. Solve either equation for x :
x + 4y = 10 ===> x = -4y + 10
Substitute into the other equation:
(-4y + 10) - 2y = -8
-6y = -18
y = 3
Solve for x :
x = -4•3 + 10
x = -12 + 10
x = -2
8. Solve the first equation for y :
-2x + y = 1 ===> y = 2x + 1
Substitute into the other equation:
4x - 2 (2x + 1) = 5
4x - 4x - 2 = 5
-2 = 5
This is false, so there is no solution.
9. Solve the second equation for x :
-x - 4y = -3 ===> x = -4y + 3
Substitute into the other equation:
2 (-4y + 3) + 8y = 6
-8y + 6 + 8y = 6
0 = 0
This means any value of x and y will satisfy these two equations, so there are infinitely many solutions.
