Algebra Examples
4x+y−2z=0 , 2x−3y+3z=9 , −6x−2y+z=0
Step 1
Choose two equations and eliminate one variable. In this case, eliminate y.
4x+y−2z=0
2x−3y+3z=9
Step 2
Step 2.1
Multiply each equation by the value that makes the coefficients of y opposite.
(3)⋅(4x+y−2z)=(3)(0)
2x−3y+3z=9
Step 2.2
Simplify.
Step 2.2.1
Simplify the left side.
Step 2.2.1.1
Simplify (3)⋅(4x+y−2z).
Step 2.2.1.1.1
Apply the distributive property.
3(4x)+3y+3(−2z)=(3)(0)
2x−3y+3z=9
Step 2.2.1.1.2
Simplify.
Step 2.2.1.1.2.1
Multiply 4 by 3.
12x+3y+3(−2z)=(3)(0)
2x−3y+3z=9
Step 2.2.1.1.2.2
Multiply −2 by 3.
12x+3y−6z=(3)(0)
2x−3y+3z=9
12x+3y−6z=(3)(0)
2x−3y+3z=9
12x+3y−6z=(3)(0)
2x−3y+3z=9
12x+3y−6z=(3)(0)
2x−3y+3z=9
Step 2.2.2
Simplify the right side.
Step 2.2.2.1
Multiply 3 by 0.
12x+3y−6z=0
2x−3y+3z=9
12x+3y−6z=0
2x−3y+3z=9
12x+3y−6z=0
2x−3y+3z=9
Step 2.3
Add the two equations together to eliminate y from the system.
| 1 | 2 | x | + | 3 | y | − | 6 | z | = | 0 | |||
| + | 2 | x | − | 3 | y | + | 3 | z | = | 9 | |||
| 1 | 4 | x | − | 3 | z | = | 9 |
Step 2.4
The resultant equation has y eliminated.
14x−3z=9
14x−3z=9
Step 3
Choose another two equations and eliminate y.
2x−3y+3z=9
−6x−2y+z=0
Step 4
Step 4.1
Multiply each equation by the value that makes the coefficients of y opposite.
(−2)⋅(2x−3y+3z)=(−2)(9)
(3)⋅(−6x−2y+z)=(3)(0)
Step 4.2
Simplify.
Step 4.2.1
Simplify the left side.
Step 4.2.1.1
Simplify (−2)⋅(2x−3y+3z).
Step 4.2.1.1.1
Apply the distributive property.
−2(2x)−2(−3y)−2(3z)=(−2)(9)
(3)⋅(−6x−2y+z)=(3)(0)
Step 4.2.1.1.2
Simplify.
Step 4.2.1.1.2.1
Multiply 2 by −2.
−4x−2(−3y)−2(3z)=(−2)(9)
(3)⋅(−6x−2y+z)=(3)(0)
Step 4.2.1.1.2.2
Multiply −3 by −2.
−4x+6y−2(3z)=(−2)(9)
(3)⋅(−6x−2y+z)=(3)(0)
Step 4.2.1.1.2.3
Multiply 3 by −2.
−4x+6y−6z=(−2)(9)
(3)⋅(−6x−2y+z)=(3)(0)
−4x+6y−6z=(−2)(9)
(3)⋅(−6x−2y+z)=(3)(0)
−4x+6y−6z=(−2)(9)
(3)⋅(−6x−2y+z)=(3)(0)
−4x+6y−6z=(−2)(9)
(3)⋅(−6x−2y+z)=(3)(0)
Step 4.2.2
Simplify the right side.
Step 4.2.2.1
Multiply −2 by 9.
−4x+6y−6z=−18
(3)⋅(−6x−2y+z)=(3)(0)
−4x+6y−6z=−18
(3)⋅(−6x−2y+z)=(3)(0)
Step 4.2.3
Simplify the left side.
Step 4.2.3.1
Simplify (3)⋅(−6x−2y+z).
Step 4.2.3.1.1
Apply the distributive property.
−4x+6y−6z=−18
3(−6x)+3(−2y)+3z=(3)(0)
Step 4.2.3.1.2
Simplify.
Step 4.2.3.1.2.1
Multiply −6 by 3.
−4x+6y−6z=−18
−18x+3(−2y)+3z=(3)(0)
Step 4.2.3.1.2.2
Multiply −2 by 3.
−4x+6y−6z=−18
−18x−6y+3z=(3)(0)
−4x+6y−6z=−18
−18x−6y+3z=(3)(0)
−4x+6y−6z=−18
−18x−6y+3z=(3)(0)
−4x+6y−6z=−18
−18x−6y+3z=(3)(0)
Step 4.2.4
Simplify the right side.
Step 4.2.4.1
Multiply 3 by 0.
−4x+6y−6z=−18
−18x−6y+3z=0
−4x+6y−6z=−18
−18x−6y+3z=0
−4x+6y−6z=−18
−18x−6y+3z=0
Step 4.3
Add the two equations together to eliminate y from the system.
| − | 4 | x | + | 6 | y | − | 6 | z | = | − | 1 | 8 | ||||
| + | − | 1 | 8 | x | − | 6 | y | + | 3 | z | = | 0 | ||||
| − | 2 | 2 | x | − | 3 | z | = | − | 1 | 8 |
Step 4.4
The resultant equation has y eliminated.
−22x−3z=−18
−22x−3z=−18
Step 5
Take the resultant equations and eliminate another variable. In this case, eliminate z.
14x−3z=9
−22x−3z=−18
Step 6
Step 6.1
Multiply each equation by the value that makes the coefficients of z opposite.
(−1)⋅(14x−3z)=(−1)(9)
−22x−3z=−18
Step 6.2
Simplify.
Step 6.2.1
Simplify the left side.
Step 6.2.1.1
Simplify (−1)⋅(14x−3z).
Step 6.2.1.1.1
Apply the distributive property.
−1(14x)−1(−3z)=(−1)(9)
−22x−3z=−18
Step 6.2.1.1.2
Multiply.
Step 6.2.1.1.2.1
Multiply 14 by −1.
−14x−1(−3z)=(−1)(9)
−22x−3z=−18
Step 6.2.1.1.2.2
Multiply −3 by −1.
−14x+3z=(−1)(9)
−22x−3z=−18
−14x+3z=(−1)(9)
−22x−3z=−18
−14x+3z=(−1)(9)
−22x−3z=−18
−14x+3z=(−1)(9)
−22x−3z=−18
Step 6.2.2
Simplify the right side.
Step 6.2.2.1
Multiply −1 by 9.
−14x+3z=−9
−22x−3z=−18
−14x+3z=−9
−22x−3z=−18
−14x+3z=−9
−22x−3z=−18
Step 6.3
Add the two equations together to eliminate z from the system.
| − | 1 | 4 | x | + | 3 | z | = | − | 9 | ||||
| + | − | 2 | 2 | x | − | 3 | z | = | − | 1 | 8 | ||
| − | 3 | 6 | x | = | − | 2 | 7 |
Step 6.4
The resultant equation has z eliminated.
−36x=−27
Step 6.5
Divide each term in −36x=−27 by −36 and simplify.
Step 6.5.1
Divide each term in −36x=−27 by −36.
−36x−36=−27−36
Step 6.5.2
Simplify the left side.
Step 6.5.2.1
Cancel the common factor of −36.
Step 6.5.2.1.1
Cancel the common factor.
−36x−36=−27−36
Step 6.5.2.1.2
Divide x by 1.
x=−27−36
x=−27−36
x=−27−36
Step 6.5.3
Simplify the right side.
Step 6.5.3.1
Cancel the common factor of −27 and −36.
Step 6.5.3.1.1
Factor −9 out of −27.
x=−9(3)−36
Step 6.5.3.1.2
Cancel the common factors.
Step 6.5.3.1.2.1
Factor −9 out of −36.
x=−9⋅3−9⋅4
Step 6.5.3.1.2.2
Cancel the common factor.
x=−9⋅3−9⋅4
Step 6.5.3.1.2.3
Rewrite the expression.
x=34
x=34
x=34
x=34
x=34
x=34
Step 7
Step 7.1
Substitute the value of x into an equation with y eliminated already.
14(34)−3z=9
Step 7.2
Solve for z.
Step 7.2.1
Simplify each term.
Step 7.2.1.1
Cancel the common factor of 2.
Step 7.2.1.1.1
Factor 2 out of 14.
2(7)34−3z=9
Step 7.2.1.1.2
Factor 2 out of 4.
2⋅732⋅2−3z=9
Step 7.2.1.1.3
Cancel the common factor.
2⋅732⋅2−3z=9
Step 7.2.1.1.4
Rewrite the expression.
7(32)−3z=9
7(32)−3z=9
Step 7.2.1.2
Combine 7 and 32.
7⋅32−3z=9
Step 7.2.1.3
Multiply 7 by 3.
212−3z=9
212−3z=9
Step 7.2.2
Move all terms not containing z to the right side of the equation.
Step 7.2.2.1
Subtract 212 from both sides of the equation.
−3z=9−212
Step 7.2.2.2
To write 9 as a fraction with a common denominator, multiply by 22.
−3z=9⋅22−212
Step 7.2.2.3
Combine 9 and 22.
−3z=9⋅22−212
Step 7.2.2.4
Combine the numerators over the common denominator.
−3z=9⋅2−212
Step 7.2.2.5
Simplify the numerator.
Step 7.2.2.5.1
Multiply 9 by 2.
−3z=18−212
Step 7.2.2.5.2
Subtract 21 from 18.
−3z=−32
−3z=−32
Step 7.2.2.6
Move the negative in front of the fraction.
−3z=−32
−3z=−32
Step 7.2.3
Divide each term in −3z=−32 by −3 and simplify.
Step 7.2.3.1
Divide each term in −3z=−32 by −3.
−3z−3=−32−3
Step 7.2.3.2
Simplify the left side.
Step 7.2.3.2.1
Cancel the common factor of −3.
Step 7.2.3.2.1.1
Cancel the common factor.
−3z−3=−32−3
Step 7.2.3.2.1.2
Divide z by 1.
z=−32−3
z=−32−3
z=−32−3
Step 7.2.3.3
Simplify the right side.
Step 7.2.3.3.1
Multiply the numerator by the reciprocal of the denominator.
z=−32⋅1−3
Step 7.2.3.3.2
Cancel the common factor of 3.
Step 7.2.3.3.2.1
Move the leading negative in −32 into the numerator.
z=−32⋅1−3
Step 7.2.3.3.2.2
Factor 3 out of −3.
z=3(−1)2⋅1−3
Step 7.2.3.3.2.3
Factor 3 out of −3.
z=3⋅−12⋅13⋅−1
Step 7.2.3.3.2.4
Cancel the common factor.
z=3⋅−12⋅13⋅−1
Step 7.2.3.3.2.5
Rewrite the expression.
z=−12⋅1−1
z=−12⋅1−1
Step 7.2.3.3.3
Multiply −12 by 1−1.
z=−12⋅−1
Step 7.2.3.3.4
Multiply 2 by −1.
z=−1−2
Step 7.2.3.3.5
Dividing two negative values results in a positive value.
z=12
z=12
z=12
z=12
z=12
Step 8
Step 8.1
Substitute the value of each known variable into one of the initial equations.
4(34)+y−2(12)=0
Step 8.2
Solve for y.
Step 8.2.1
Simplify 4(34)+y−2(12).
Step 8.2.1.1
Simplify each term.
Step 8.2.1.1.1
Cancel the common factor of 4.
Step 8.2.1.1.1.1
Cancel the common factor.
4(34)+y−2(12)=0
Step 8.2.1.1.1.2
Rewrite the expression.
3+y−2(12)=0
3+y−2(12)=0
Step 8.2.1.1.2
Cancel the common factor of 2.
Step 8.2.1.1.2.1
Factor 2 out of −2.
3+y+2(−1)12=0
Step 8.2.1.1.2.2
Cancel the common factor.
3+y+2⋅−112=0
Step 8.2.1.1.2.3
Rewrite the expression.
3+y−1=0
3+y−1=0
3+y−1=0
Step 8.2.1.2
Subtract 1 from 3.
y+2=0
y+2=0
Step 8.2.2
Subtract 2 from both sides of the equation.
y=−2
y=−2
y=−2
Step 9
The solution to the system of equations can be represented as a point.
(34,−2,12)
Step 10
The result can be shown in multiple forms.
Point Form:
(34,−2,12)
Equation Form:
x=34,y=−2,z=12
