For this problem, I would put all the equations in slope-intercept form (y = mx + b) and graph each one.
1) 3x + 2y = 6
Subtract 3x from both sides of the equation.
2y = -3x + 6
Divide all terms by 2.
y = -3/2x + 3
The graph of this line has a y-intercept of 3 and a negative slope of 3/2. Note that the line has a y-intercept of three which means it crosses the coordinate (0,3).
Solution: (0,3)
2) -5x + y = -10
Add -5x to both sides of the equation.
y = 5x - 10
The graph of this line has a y-intercept of -10 and a slope of 5. If the line's y-intercept is -10 and the slope is positive 5, then the line will have to rise 5 units and run 1 unit to the left.
(0, -10) → (1, -5)
Solution: (1, -5)
3) x - 4y = 8
Subtract x from both sides of the equation.
-4y = -x + 8
Divide all terms by -4.
y = x/4 - 2
y = 1/4x - 2
The line has a y-intercept of -2 and a slope of 1/4. If the line's y-intercept is -2 and the slope is 1/4, then the line will have to rise 1 unit and run 4 units to the left.
(0, -2) → (4, -1)
Solution: (4, -1)
4) By process of elimination, we'll know that the equation of -6x - 5y = 30 should have the solution of (0, -6). However, it's good to check the answer.
-6x - 5y = 30
Add -6x to both sides of the equation.
-5y = 6x + 30
Divide all terms by -5.
y = -6/5x - 6
The y-intercept is (0, -6) and the slope is 6 units down over 5 units to the right.
Solution: (0, -6)