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QuestionSolve the system of equations: -2x - 2y + 3z = -19, 3x - 4y - 2z = -28, 6x + 3y - 4z = 21.

Studdy Solution

STEP 1

Assumptions1. The system of equations is linear. . The system of equations has a unique solution.

STEP 2

The system of equations is given by{−2x−2y+z=−19x−4y−2z=−286x+y−4z=21\left\{\begin{array}{l}-2 x-2 y+ z=-19 \\ x-4 y-2 z=-28 \\6 x+ y-4 z=21\end{array}\right.

STEP 3

We can solve this system of equations using the method of substitution or elimination. Here, we will use the method of elimination.

STEP 4

First, let's multiply the first equation by3 and the second equation by2 to make the coefficients of x in the first two equations equal and opposite, so they can be eliminated when summed.
{−6x−6y+9z=−576x−8y−4z=−566x+3y−4z=21\left\{\begin{array}{l}-6 x-6 y+9 z=-57 \\6 x-8 y-4 z=-56 \\6 x+3 y-4 z=21\end{array}\right.

STEP 5

Now, add the first and second equations to eliminate x.
−14y+5z=−113-14y +5z = -113

STEP 6

Next, multiply the first equation by2 and the third equation by3 to make the coefficients of x in the first and third equations equal and opposite.
{−12x−12y+18z=−1146x−8y−4z=−5618x+9y−12z=63\left\{\begin{array}{l}-12 x-12 y+18 z=-114 \\6 x-8 y-4 z=-56 \\18 x+9 y-12 z=63\end{array}\right.

STEP 7

Now, add the first and third equations to eliminate x.
−3y+6z=−51-3y +6z = -51

STEP 8

We now have a system of two equations with two variables, y and z.{−14y+5z=−113−3y+6z=−51\left\{\begin{array}{l}-14y +5z = -113 \\ -3y +6z = -51\end{array}\right.

STEP 9

Multiply the second equation by14 and the first equation by3 to make the coefficients of y in both equations equal and opposite.
{−42y+15z=−33942y−84z=−714\left\{\begin{array}{l}-42y +15z = -339 \\42y -84z = -714\end{array}\right.

STEP 10

Add the two equations to eliminate y.
−69z=−1053-69z = -1053

STEP 11

olve for z.
z=−1053−69=15.26z = \frac{-1053}{-69} =15.26

STEP 12

Substitute z =15.26 into the equation -y +6z = -51 to solve for y.
−y+6(15.26)=−51-y +6(15.26) = -51

STEP 13

olve for y.
y=−51−6(15.26)−3=7.13y = \frac{-51 -6(15.26)}{-3} =7.13

STEP 14

Substitute y =7.13 and z =.26 into the equation -2x -2y +3z = -19 to solve for x.
−2x−2(7.13)+3(.26)=−19-2x -2(7.13) +3(.26) = -19

STEP 15

olve for x.
x=−19+2(7.13)−3(15.26)−2=−4.87x = \frac{-19 +2(7.13) -3(15.26)}{-2} = -4.87The solution to the system of equations is x = -4.87, y =7.13, z =15.26.

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