How do you solve the simultaneous equations #x+y+z=-2#, #2x+5y+2z=-10#, #-x+6y-3z=-16# ?

Answer 1

#(x, y, z) = (-2, -2, 2)#

Given:

#{ (x+y+z=-2),(2x+5y+2z=-10),(-x+6y-3z=-16) :}#

Subtracting twice the first equation from the second, we get:

#3y = -6#
Dividing both sides by #3# we find:
#y = -2#

Adding the first and third equation together, we get:

#7y-2z = -18#
Substituting #y=-2# into this equation, we get:
#-14-2z = -18#
Add #14# to both sides to get:
#-2z = -4#
Divide both sides by #-2# to get:
#z = 2#
Then putting #y=-2# and #z = 2# in the first equation, we find:
#x-color(red)(cancel(color(black)(2)))+color(red)(cancel(color(black)(2)))=-2#

Hence:

#x = -2#
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Answer 2

Use the 3 equations to write an Augmented Matrix and then perform elementary row operations until you obtain an identity matrix.

Write the augmented matrix:

#[ (1,1,1,|,-2), (2,5,2,|,-10), (-1,6,-3,|,-16) ]#

Perform elementary row operations.

#-2R_1+R_2toR_2#

#[ (1,1,1,|,-2), (0,3,0,|,-6), (-1,6,-3,|,-16) ]#

#R_2/3#

#[ (1,1,1,|,-2), (0,1,0,|,-2), (-1,6,-3,|,-16) ]#

#R_1+R_3toR_3#

#[ (1,1,1,|,-2), (0,1,0,|,-2), (0,7,-2,|,-18) ]#

#-7R_2+R_3toR_3#

#[ (1,1,1,|,-2), (0,1,0,|,-2), (0,0,-2,|,-4) ]#

#R_3/-2#

#[ (1,1,1,|,-2), (0,1,0,|,-2), (0,0,1,|,2) ]#

#R_1-R_2toR_1#

#[ (1,0,1,|,0), (0,1,0,|,-2), (0,0,1,|,2) ]#

#R_1-R_3toR_1#

#[ (1,0,0,|,-2), (0,1,0,|,-2), (0,0,1,|,2) ]#

We have obtained an identity matrix and the right column contains the solution set:

#x = -2, y = -2, and z = 2#
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Answer 3

To solve the simultaneous equations:

  1. Use the first equation to express one variable in terms of the others. For example, express ( x ) in terms of ( y ) and ( z ).

  2. Substitute the expression for ( x ) into the other two equations.

  3. This will create two equations with two variables each.

  4. Solve the resulting equations using any method of solving systems of linear equations, such as substitution, elimination, or matrix methods.

  5. Once you find the values of ( y ) and ( z ), substitute them back into any of the original equations to find the value of ( x ).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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