Solve the following system of equations algebraically. Verify the solution by using matrices.
8x-2y=5
-12x+3y=7
you could eliminate the x by doing some judicious multiplying:
24x-6y = 15
-24x+6y = 14
Now add the equations to get
0y = 29
BZZZT! No solution
Thank you
To solve this system of equations algebraically, we can use the method of substitution or elimination. Let's use the method of elimination.
We have the system of equations:
8x - 2y = 5 ...(1)
-12x + 3y = 7 ...(2)
To eliminate one of the variables, we need to multiply one or both equations by a constant so that the coefficients of one of the variables will cancel each other out when we add or subtract the equations.
Let's start by multiplying equation (1) by 3 and equation (2) by 2 to eliminate the 'y'.
Multiplying equation (1) by 3:
(3)(8x - 2y) = (3)(5)
24x - 6y = 15 ...(3)
Multiplying equation (2) by 2:
(2)(-12x + 3y) = (2)(7)
-24x + 6y = 14 ...(4)
Now, we can subtract equation (4) from equation (3) to eliminate 'y':
(24x - 6y) - (-24x + 6y) = 15 - 14
24x - 6y + 24x - 6y = 1
48x - 12y = 1 ...(5)
Now, let's solve equation (5) for 'x':
48x - 12y = 1
48x = 12y + 1
x = (12y + 1) / 48
x = (3y + 1) / 12 ...(6)
Now, substitute the value of 'x' from equation (6) into either equation (1) or (2). Let's use equation (1):
8x - 2y = 5
8[(3y + 1) / 12] - 2y = 5
(24y + 8) / 12 - 2y = 5
(24y + 8 - 24y) / 12 = 5
8 / 12 = 5
2 / 3 = 5
2 = 3 * 5
2 = 15
This equation is false, which means there is no solution to the system of equations.
To verify this result using matrices, we can write the system of equations in augmented matrix form and row-reduce it.
The augmented matrix for the system is:
[8 -2 | 5]
[-12 3 | 7]
Using row operations, let's row-reduce the augmented matrix:
R2 = R2 + 3/2 * R1 (Multiply row 1 by 3/2 and add it to row 2)
[-18 0 | 17]
[0 0 | 0 ]
The row-reduced augmented matrix shows that the last row is all zeros, which means there are infinitely many solutions to the system of equations.
Therefore, the system of equations is inconsistent, and there is no unique solution.