Studdy AI Logo
Math  /  Algebra

QuestionUse any convenient method to solve the following system of equations. If the system is dependent, express the solution set in terms of one of the variables. Leave all fractional answers in fraction form. {−3x+3y+6z=−12x+7y−z=274x+4y−7z=39\left\{\begin{array}{rr} -3 x+3 y+6 z= & -12 \\ x+7 y-z= & 27 \\ 4 x+4 y-7 z= & 39 \end{array}\right.

Studdy Solution

STEP 1

1. We are given a system of three linear equations with three variables: xx, yy, and zz.
2. The system can be solved using methods such as substitution, elimination, or matrix operations.
3. The solution could be a single point, infinitely many solutions (dependent system), or no solution (inconsistent system).

STEP 2

1. Choose a method to solve the system of equations.
2. Simplify the system and solve for one variable.
3. Substitute back to find the other variables.
4. Check the solution or determine if the system is dependent.

STEP 3

Let's use the elimination method to simplify the system. Our goal is to eliminate one variable from two pairs of equations.

STEP 4

First, let's eliminate xx from the first and second equations. Multiply the second equation by 3 to align the coefficients of xx:
3(x+7y−z)=3(27)3(x + 7y - z) = 3(27)
This gives:
3x+21y−3z=813x + 21y - 3z = 81
Now add this equation to the first equation:
(−3x+3y+6z)+(3x+21y−3z)=−12+81(-3x + 3y + 6z) + (3x + 21y - 3z) = -12 + 81
Simplify:
24y+3z=6924y + 3z = 69

STEP 5

Next, eliminate xx from the second and third equations. Multiply the second equation by 4:
4(x+7y−z)=4(27)4(x + 7y - z) = 4(27)
This gives:
4x+28y−4z=1084x + 28y - 4z = 108
Subtract the third equation from this result:
(4x+28y−4z)−(4x+4y−7z)=108−39(4x + 28y - 4z) - (4x + 4y - 7z) = 108 - 39
Simplify:
24y+3z=6924y + 3z = 69

STEP 6

Notice that both reduced equations are identical, 24y+3z=6924y + 3z = 69. This indicates that the system is dependent. We can express the solution set in terms of one variable, say zz.
Solve for yy in terms of zz:
24y=69−3z24y = 69 - 3z
y=69−3z24y = \frac{69 - 3z}{24}

STEP 7

Substitute y=69−3z24y = \frac{69 - 3z}{24} into one of the original equations to express xx in terms of zz. Let's use the second equation:
x+7(69−3z24)−z=27x + 7\left(\frac{69 - 3z}{24}\right) - z = 27
Simplify:
x+483−21z24−z=27x + \frac{483 - 21z}{24} - z = 27
Multiply through by 24 to clear the fraction:
24x+483−21z−24z=64824x + 483 - 21z - 24z = 648
24x=648−483+45z24x = 648 - 483 + 45z
24x=165+45z24x = 165 + 45z
x=165+45z24x = \frac{165 + 45z}{24}

STEP 8

The solution set can be expressed in terms of zz:
x=165+45z24,y=69−3z24,z=zx = \frac{165 + 45z}{24}, \quad y = \frac{69 - 3z}{24}, \quad z = z
This confirms the system is dependent, and the solution set is expressed in terms of zz.

Was this helpful?

Studdy solves anything!

banner

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord