Answer:
(-6, 4/3, 7)
Explanation:
To solve the system of equations, we can use elimination or substitution. Here, we will use elimination:
First, we can eliminate z by adding the first and third equations:
x - 3y - 5z = 7
6x + 5y - 6z = 85
7x + 2y = 92
Next, we can use the second equation to eliminate z from the second equation:
6y + z = 27
-6z from both sides:
6y = -6z + 27
Divide both sides by 6:
y = -z/6 + 9/2
Now, we can substitute y = -z/6 + 9/2 into the equation 7x + 2y = 92:
7x + 2(-z/6 + 9/2) = 92
7x - z/3 + 9 = 92
7x - z/3 = 83
Multiplying both sides by 3 to eliminate the fraction:
21x - z = 249
We now have two equations:
7x + 2y = 92
21x - z = 249
To eliminate x, we can multiply the first equation by 3 and subtract it from the second equation:
21x - z = 249
- (3 * (7x + 2y = 92))
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-13x - 6z = -215
Solving this equation for z, we get:
-13x - 6z = -215
6z = 13x + 215
z = (13/6)x + 35 + 1/3
Now we can substitute this value of z into one of the equations to solve for y. Let's use the second equation:
6y + z = 27
6y + [(13/6)x + 35 + 1/3] = 27
6y + (13/6)x + 35 + 1/3 - 27 = 0
6y + (13/6)x + 8 + 1/3 = 0
6y + (13/6)x = -25/3
y = (-13/36)x - (25/18)
Finally, we can substitute these values for x, y, and z into one of the original equations to check our work. Let's use the first equation:
x - 3y - 5z = 7
x - 3[(-13/36)x - (25/18)] - 5[(13/6)x + 35 + 1/3] = 7
x + (13/12)x + (25/6) - (65/6)x - 175 - (5/3) = 7
-51x/12 - 215/6 = 7
-17x/4 - 35/2 = 7
-17x/4 = 21/2
x = -6
Substituting this value of x into the expressions we found for y and z, we get:
y = (-13/36)x - (25/18) = (13/6) - (25/18) = 4/3
z = (13/6)x + 35 + 1/3 = -(13/6)*6 + 35 + 1/3 = 7