Solve the system of equations by the elimination method: 3x-2y+3=25
2x-7y-4=12
To solve the system of equations by the elimination method, we need to eliminate one of the variables by multiplying one or both of the equations by a constant so that the coefficients of that variable have the same absolute value, but opposite signs.
Let's start by eliminating the y variable. We can do this by multiplying the first equation by 7 and the second equation by 2.
7(3x - 2y + 3) = 7(25)
2(2x - 7y - 4) = 2(12)
Simplifying both equations:
21x - 14y + 21 = 175
4x - 14y - 8 = 24
Now we have a new system of equations:
21x - 14y = 154
4x - 14y = 32
Subtracting the second equation from the first equation:
(21x - 14y) - (4x - 14y) = 154 - 32
21x - 14y - 4x + 14y = 122
(21x - 4x) + (-14y + 14y) = 122
17x = 122
x = 122 / 17
x = 7.1765 (rounded to four decimal places)
Now substitute the value of x into one of the original equations to solve for y. Let's use the first equation:
3x - 2y + 3 = 25
3(7.1765) - 2y + 3 = 25
21.5295 - 2y + 3 = 25
-2y = 25 - 21.5295 - 3
-2y = 0.4705
y = 0.4705 / -2
y = -0.23525 (rounded to five decimal places)
Therefore, the solution to the system of equations is x = 7.1765 and y = -0.23525.