a. 2x + 3y = 7
2y = 12x - 2
To solve this system of equations, we can begin by isolating one of the variables. Let's first isolate y in the second equation:
2y = 12x - 2
y = 6x - 1
Now we can substitute this expression for y into the first equation:
2x + 3(6x - 1) = 7
2x + 18x - 3 = 7
20x = 10
x = 1/2
Now that we know the value of x, we can substitute it back into the equation y = 6x - 1 to find the value of y:
y = 6(1/2) - 1
y = 3/2 - 1
y = -1/2
So the solution to this system of equations is (x, y) = (1/2, -1/2).
b. 2x - 9y = 12
3x + 18y = 18
To solve this system of equations, we can begin by using the first equation to solve for one of the variables. Let's solve for x:
2x - 9y = 12
2x = 9y + 12
x = (9y + 12)/2
Now we can substitute this expression for x into the second equation:
3((9y + 12)/2) + 18y = 18
(27y + 36) + 18y = 18
45y = -18
y = -4/5
Now we can substitute this value of y back into the equation x = (9y + 12)/2 to find the value of x:
x = (9(-4/5) + 12)/2
x = (3 + 12)/2
x = 15/2
So the solution to this system of equations is (x, y) = (15/2, -4/5).