What is the solution of the system of equations?
{-3x - 4y - 3z = -7
{2x - 6y + 2z = 3
{5x - 2y + 5z = 9
A. (5,-2,7)
B. (-5,2,7)
C.(5,2,-7)
D. No solution
I get no solution, since row2-row1 = row3 but with a different right-hand side
To find the solution of the system of equations, we can use the method of Gaussian elimination to row-reduce the augmented matrix.
The augmented matrix for the system of equations is:
[-3 -4 -3 | -7]
[ 2 -6 2 | 3]
[ 5 -2 5 | 9]
First, we will perform row operations to eliminate the coefficients below the main diagonal. Start by multiplying the first row by 2 and adding it to the second row:
[ -3 -4 -3 | -7]
[ 0 -14 -4 | -1]
[ 5 -2 5 | 9]
Next, multiply the first row by -5 and add it to the third row:
[ -3 -4 -3 | -7]
[ 0 -14 -4 | -1]
[ 0 18 20 | -26]
To further simplify the matrix, we can divide the second row by -14:
[ -3 -4 -3 | -7]
[ 0 1 2/7 | 1/14]
[ 0 18 20 | -26]
Next, multiply the second row by 18 and add it to the third row:
[ -3 -4 -3 | -7]
[ 0 1 2/7 | 1/14]
[ 0 0 20/7 | -13/14]
To simplify the matrix further, we can divide the third row by 20/7:
[ -3 -4 -3 | -7]
[ 0 1 2/7 | 1/14]
[ 0 0 1 | -13/20]
From the row-reduced matrix, we can see that the variables x, y, and z have the following values:
x = -7
y = 1/14
z = -13/20
So, the solution to the system of equations is (-7, 1/14, -13/20).
None of the given answer choices match the solution exactly, so the correct option would be D. No solution.
To find the solution to the system of equations, we can use the method of substitution or elimination. Let's use the method of substitution:
1. Begin by rearranging one equation to solve for one variable in terms of the others.
Let's start with the first equation: -3x - 4y - 3z = -7.
Solving for x, we have: -3x = -7 + 4y + 3z -> x = (7 + 4y + 3z) / -3.
2. Substitute the expression for x in the other equations.
Substituting in the second equation: 2((7 + 4y + 3z) / -3) - 6y + 2z = 3,
Simplifying, we get: (14 + 8y + 6z) / -3 - 6y + 2z = 3.
3. Continue this process for the third equation.
Substituting in the third equation: 5((7 + 4y + 3z) / -3) - 2y + 5z = 9,
Simplifying, we get: (35 + 20y + 15z) / -3 - 2y + 5z = 9.
4. Multiply each equation by -3 to eliminate the fractions.
The equations become:
14 + 8y + 6z + 18y - 6z = -9,
35 + 20y + 15z + 6y - 15z = -27.
5. Combine like terms.
Simplifying, we get:
22y = -23,
26y = -62.
6. Solve for y.
From the first equation, we have: y = -23/22,
From the second equation, we have: y = -62/26.
Since the equation for y leads to different solutions, there is no consistent solution for the system of equations. Thus, the correct answer is D. No solution.