Find the Variables

Problem

a*[[1],[−1],[8]]+b*[[2],[1],[7]]+d*[[1],[3],[−4]]+g*[[3],[−4],[36]]=[[10],[−15],[131]]

Solution

1. Write the system of linear equations by multiplying the scalars into the vectors and summing the components.

a+2*b+d+3*g=10

−a+b+3*d−4*g=−15

8*a+7*b−4*d+36*g=131

2. Eliminate a by adding the first equation to the second equation.

(a−a)+(2*b+b)+(d+3*d)+(3*g−4*g)=10−15

3*b+4*d−g=−5

3. Eliminate a from the third equation by multiplying the first equation by 8 and subtracting it from the third.

(8*a−8*a)+(7*b−16*b)+(−4*d−8*d)+(36*g−24*g)=131−80

−9*b−12*d+12*g=51

4. Simplify the new third equation by dividing all terms by -3.

3*b+4*d−4*g=−17

5. Eliminate b and d by subtracting the simplified third equation from the equation found in step 2.

(3*b−3*b)+(4*d−4*d)+(−g−(−4*g))=−5−(−17)

3*g=12

g=4

6. Substitute g=4 back into the equation 3*b+4*d−g=−5 to find a relationship between b and d

3*b+4*d−4=−5

3*b+4*d=−1

7. Identify the system as underdetermined because there are four variables and only three equations. Express b d and a in terms of a free variable. Let d be the free variable.

3*b=−1−4*d

b=(−1−4*d)/3

8. Substitute b and g back into the first equation to solve for a in terms of d

a+2*((−1−4*d)/3)+d+3*(4)=10

a−2/3−(8*d)/3+d+12=10

a−(5*d)/3+34/3=10

a=(5*d)/3−4/3

Final Answer

a=(5*d−4)/3,b=(−4*d−1)/3,g=4,d∈ℝ

Want more problems? Check here!