Question 208209
you have 3 equations in 3 unknowns.
you want to reduce that to 2 equations in 2 unknowns
you then reduce that to 1 equation in one unknown
working your way back up the chain...
you then solve for the first unknown
you then solve for the second unknown
you then solve for the third unknown.
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your equations are:
x - y + 5z = 2 = equation 1
4x - 3y + 5z = 3 = equation 2
3x - 2y + 4z = 1 = equation 3
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you can eliminate z first.
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you subtract equation 1 from equation 2 to get:
3x - 2y = 1 = equation 4
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you multiply equation 1 by 4 to get:
4x - 4y + 20z = 8 = equation 5
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you multiply equation 3 by 5 to get:
15x - 10y + 20z = 5 = equation 6
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you subtract equation 5 from equation 6 to get:
11x - 6y = -3 = equation 7
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you now have 2 equations in 2 unknowns.
those are equation 4 and equation 7 shown below:
3x - 2y = 1 = equation 4
11x - 6y = -3 = equation 7
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next you will eliminate the y to get one equation in one unknown.,
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you multiply equation 4 by 3 to get:
9x - 6y = 3 = equation 8
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you subtract equation 8 from equation 7 to get:
2x = -6
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you now have one equation in one unknown which you solve to get:
x = -3
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you now go back to equation 4 or equation 7 and solve for y.
let's use equation 7
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11x - 6y = -3 = equation 7
replace x with -3 to get:
11 *(-3) - 6y = -3
-33 - 6y = -3
6y = -30
y = -5
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you now have:
x = -3
y = -5
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go back to equation 1 or 2 or 3 and replace x with -3 and replace y with -5 to solve for z.
we'll use equation 2 as shown below:
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4x - 3y + 5z = 3 = equation 2
replace x with -3 and y with -5 to get:
4 * (-3) - 3 * (-5) + 5z = 3
-12 + 15 + 5z = 3
3 + 5z = 3
5z = 0
z = 0
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you now have:
x = -3
y = -5
z = 0
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your equation should be solved and you confirm that by replacing x with -3 and y with -5 and z with 0 in all 3 of your equations to make sure that the equations are true with those values in them for x,y,z.
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your original equations are:
x - y + 5z = 2 = equation 1
4x - 3y + 5z = 3 = equation 2
3x - 2y + 4z = 1 = equation 3
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equation 1 becomes -3 + 5 + 0 = 2 which becomes 2 = 2 which is true.
equation 2 becomes -12 + 15 + 0 = 3 which becomes 3 = 3 which is true.
equation 3 becomes -9 + 10 + 0 = 1 which becomes 1 = 3 which is true.
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all 3 equations are true so the values for x,y,z satisfy all of them simultaneously and you are done.
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