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Math Question - Solve for Three Unknowns











































(1 pt) Solve
5x-6y+2z=2
x-y+z=3
4x-3y=1
x = , y = , z = .

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Mar 24, 2011
Solve for Three Unknowns
by: Staff

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Part II

Solve for x by substituting the numerical value of y in the following equation:

x = .25 + (.75)*y

x = .25 + (.75)*(2.7142857142857143)

x = .25 + 2.0357142857142858

x = 2.2857142857142857

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Solve for z by substituting the numerical value of y in the following equation:

z = 2.75 + .25*y

z = 2.75 + .25*2.7142857142857143

z = 3.428571428571428571


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The final answer is:

x = 2.2857142857142857

y = 2.7142857142857143

z = 3.4285714285714286

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The last step is to check the answers by substituting the values of x, y, and z in the original three equations:

5x - 6y + 2z = 2

5*2.2857142857142857 – 6*2.7142857142857143 + 2*3.4285714285714286 = 2, correct


x – y + z = 3

2.2857142857142857 - 2.7142857142857143 + 3.4285714285714286 = 3, correct


4x - 3y = 1

4*2.2857142857142857 – 3*2.7142857142857143 = 1, correct

This final check verified that the values of x, y, and z are correct solutions to the three simultaneous equations.



Thanks for writing.


Staff
www.solving-math-problems.com


Mar 24, 2011
Solve for Three Unknowns
by: Staff


Part I

The question:
(1 pt) Solve

5x - 6y + 2z = 2

x – y + z = 3

4x - 3y = 1

x = , y = , z = .


The answer:

Three simultaneous equations can be solved using (1) Matrices and Row Reduction, or (2) Substitution.

Since you did not specify the use of matrices, I presume you would like to solve the equation using substitution.

Using this method, the strategy is to use substitution to reduce the number of equations from three equations to two equations, and then reduce the two equations to one equation.

Original three simultaneous equations:

5x - 6y + 2z = 2

x – y + z = 3

4x - 3y = 1

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Solve for x using equation 3

4x - 3y = 1

Add 3y to each side of the equation

4x - 3y + 3y = 1 + 3y

4x + 0 = 1 + 3y

4x = 1 + 3y


Divide each side of the equation by 4

4x = 1 + 3y

4x/4 = (1 + 3y)/4

x = (1/4) + (3/4)*y

x = .25 + (.75)*y


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Substitute (1/4) + (3/4)*y for the value of x into equation 2

x – y + z = 3

(1/4) + (3/4)*y – y + z = 3

Combine terms

(1/4) + (3/4)*y – y + z = 3

(1/4) - (1/4)*y + z = 3

Solve for z

(1/4) - (1/4)*y + z = 3

Subtract ¼ from each side of the equation

(1/4) – (1/4) - (1/4)*y + z = 3 – (1/4)

0 - (1/4)*y + z = 3 – (1/4)

-(1/4)*y + z = 3 – (1/4)

-.25*y + z = 2.75

Add .25*y from each side of the equation

-.25*y + .25*y + z = 2.75 + .25*y

0 + z = 2.75 + .25*y

z = 2.75 + .25*y


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Substitute the values of x [x = .25 + (.75)*y]
and z [z = 2.75 + .25*y] in equation 1


5x - 6y + 2z = 2

5*[.25 + (.75)*y] - 6y + 2*[2.75 + .25*y] = 2

There is now only 1 equation with 1 unknown.

Solve for y

5*[.25 + (.75)*y] - 6y + 2*[2.75 + .25*y] = 2

Eliminate the [ ] brackets using the distributive law:

1.25 + 3.75y - 6y + 5.5 + .50y = 2

Combine terms:

6.75 - 1.75y = 2

Add 1.75y to each side of the equation

6.75 - 1.75y + 1.75y = 2 + 1.75y

6.75 + 0 = 2 + 1.75y

6.75 = 2 + 1.75y

Subtract 2 from each side of the equation

6.75 - 2 = 2 - 2 + 1.75y

6.75 - 2 = 0 + 1.75y

6.75 - 2 = 1.75y

4.75 = 1.75y

Divide each side of the equation by 1.75

4.75/1.75 = 1.75y/1.75

4.75/1.75 = y*(1.75/1.75)

4.75/1.75 = y*(1)

4.75/1.75 = y

4.75/1.75 = y

2.7142857142857143 = y

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