How do you solve the simultaneous equations #x+y+z=-2#, #2x+5y+2z=-10#, #-x+6y-3z=-16# ?
Given:
Subtracting twice the first equation from the second, we get:
Adding the first and third equation together, we get:
Hence:
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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.
#[ (1,1,1,|,-2), (0,3,0,|,-6), (-1,6,-3,|,-16) ]#
#[ (1,1,1,|,-2), (0,1,0,|,-2), (-1,6,-3,|,-16) ]#
#[ (1,1,1,|,-2), (0,1,0,|,-2), (0,7,-2,|,-18) ]#
#[ (1,1,1,|,-2), (0,1,0,|,-2), (0,0,-2,|,-4) ]#
#[ (1,1,1,|,-2), (0,1,0,|,-2), (0,0,1,|,2) ]#
#[ (1,0,1,|,0), (0,1,0,|,-2), (0,0,1,|,2) ]#
#[ (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:
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To solve the simultaneous equations:
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Use the first equation to express one variable in terms of the others. For example, express ( x ) in terms of ( y ) and ( z ).
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Substitute the expression for ( x ) into the other two equations.
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This will create two equations with two variables each.
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Solve the resulting equations using any method of solving systems of linear equations, such as substitution, elimination, or matrix methods.
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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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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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