SOLUTION: solve the system of equations using elimination or substitition
x-2y+4z=-19
2x+y-3z=14
3x+y+2z=5
Algebra.Com
Question 347567: solve the system of equations using elimination or substitition
x-2y+4z=-19
2x+y-3z=14
3x+y+2z=5
Answer by haileytucki(390) (Show Source): You can put this solution on YOUR website!
x-2y+4z=-19_2x+y-3z=14_3x+y+2z=5
Move all terms not containing x to the right-hand side of the equation.
x=2y-4z-19_2x+y-3z=14_3x+y+2z=5
Replace all occurrences of x with the solution found by solving the last equation for x. In this case, the value substituted is 2y-4z-19.
x=2y-4z-19_2(2y-4z-19)+y-3z=14_3x+y+2z=5
Replace all occurrences of x with the solution found by solving the last equation for x. In this case, the value substituted is 2y-4z-19.
x=2y-4z-19_2(2y-4z-19)+y-3z=14_3(2y-4z-19)+y+2z=5
Multiply 2 by each term inside the parentheses.
x=2y-4z-19_4y-8z-38+y-3z=14_3(2y-4z-19)+y+2z=5
Since 4y and y are like terms, add y to 4y to get 5y.
x=2y-4z-19_5y-8z-38-3z=14_3(2y-4z-19)+y+2z=5
Since -8z and -3z are like terms, subtract 3z from -8z to get -11z.
x=2y-4z-19_5y-11z-38=14_3(2y-4z-19)+y+2z=5
Multiply 3 by each term inside the parentheses.
x=2y-4z-19_5y-11z-38=14_6y-12z-57+y+2z=5
Since 6y and y are like terms, add y to 6y to get 7y.
x=2y-4z-19_5y-11z-38=14_7y-12z-57+2z=5
Since -12z and 2z are like terms, subtract 2z from -12z to get -10z.
x=2y-4z-19_5y-11z-38=14_7y-10z-57=5
Move all terms not containing y to the right-hand side of the equation.
x=2y-4z-19_5y=11z+38+14_7y-10z-57=5
Add 14 to 38 to get 52.
x=2y-4z-19_5y=11z+52_7y-10z-57=5
Divide each term in the equation by 5.
x=2y-4z-19_(5y)/(5)=(11z)/(5)+(52)/(5)_7y-10z-57=5
Simplify the left-hand side of the equation by canceling the common factors.
x=2y-4z-19_y=(11z)/(5)+(52)/(5)_7y-10z-57=5
Combine the numerators of all expressions that have common denominators.
x=2y-4z-19_y=(11z+52)/(5)_7y-10z-57=5
Replace all occurrences of y with the solution found by solving the last equation for y. In this case, the value substituted is ((11z+52))/(5).
x=2y-4z-19_y=(11z+52)/(5)_7((11z+52)/(5))-10z-57=5
Remove the parentheses around the expression 11z+52.
x=2y-4z-19_y=(11z+52)/(5)_7((11z+52)/(5))-10z-57=5
Divide each term in the numerator by the denominator.
x=2y-4z-19_y=(11z)/(5)+(52)/(5)_7((11z+52)/(5))-10z-57=5
Remove the parentheses around the expression 11z+52.
x=2y-4z-19_y=(11z)/(5)+(52)/(5)_7((11z+52)/(5))-10z-57=5
Divide each term in the numerator by the denominator.
x=2y-4z-19_y=(11z)/(5)+(52)/(5)_7((11z)/(5)+(52)/(5))-10z-57=5
Combine the numerators of all expressions that have common denominators.
x=2y-4z-19_y=(11z)/(5)+(52)/(5)_7((11z+52)/(5))-10z-57=5
Divide each term in the numerator by the denominator.
x=2y-4z-19_y=(11z)/(5)+(52)/(5)_7((11z)/(5)+(52)/(5))-10z-57=5
Multiply 7 by each term inside the parentheses.
x=2y-4z-19_y=(11z)/(5)+(52)/(5)_(77z+364)/(5)-10z-57=5
Divide each term in the numerator by the denominator.
x=2y-4z-19_y=(11z)/(5)+(52)/(5)_(77z)/(5)+(364)/(5)-10z-57=5
To add fractions, the denominators must be equal. The denominators can be made equal by finding the least common denominator (LCD). In this case, the LCD is 5. Next, multiply each fraction by a factor of 1 that will create the LCD in each of the fractions.
x=2y-4z-19_y=(11z)/(5)+(52)/(5)_(77z)/(5)-10z-57*(5)/(5)+(364)/(5)=5
Complete the multiplication to produce a denominator of 5 in each expression.
x=2y-4z-19_y=(11z)/(5)+(52)/(5)_(77z)/(5)-10z-(285)/(5)+(364)/(5)=5
Combine the numerators of all fractions that have common denominators.
x=2y-4z-19_y=(11z)/(5)+(52)/(5)_(77z)/(5)-10z+(-285+364)/(5)=5
Add 364 to -285 to get 79.
x=2y-4z-19_y=(11z)/(5)+(52)/(5)_(77z)/(5)-10z+(79)/(5)=5
Combine the numerators of all expressions that have common denominators.
x=2y-4z-19_y=(11z)/(5)+(52)/(5)_(77z+79)/(5)-10z=5
Multiply each term by a factor of 1 that will equate all the denominators. In this case, all terms need a denominator of 5.
x=2y-4z-19_y=(11z)/(5)+(52)/(5)_(77z+79)/(5)-10z*(5)/(5)=5
Multiply the expression by a factor of 1 to create the least common denominator (LCD) of 5.
x=2y-4z-19_y=(11z)/(5)+(52)/(5)_(77z+79)/(5)-(10z*5)/(5)=5
Multiply 10z by 5 to get 50z.
x=2y-4z-19_y=(11z)/(5)+(52)/(5)_(77z+79)/(5)-(50z)/(5)=5
The numerators of expressions that have equal denominators can be combined. In this case, ((77z+79))/(5) and -((50z))/(5) have the same denominator of 5, so the numerators can be combined.
x=2y-4z-19_y=(11z)/(5)+(52)/(5)_((77z+79)-(50z))/(5)=5
Simplify the numerator of the expression.
x=2y-4z-19_y=(11z)/(5)+(52)/(5)_(77z+79-50z)/(5)=5
Since 77z and -50z are like terms, add -50z to 77z to get 27z.
x=2y-4z-19_y=(11z)/(5)+(52)/(5)_(27z+79)/(5)=5
Multiply each term in the equation by 5.
x=2y-4z-19_y=(11z)/(5)+(52)/(5)_(27z+79)/(5)*5=5*5
Simplify the left-hand side of the equation by canceling the common factors.
x=2y-4z-19_y=(11z)/(5)+(52)/(5)_27z+79=5*5
Multiply 5 by 5 to get 25.
x=2y-4z-19_y=(11z)/(5)+(52)/(5)_27z+79=25
Since 79 does not contain the variable to solve for, move it to the right-hand side of the equation by subtracting 79 from both sides.
x=2y-4z-19_y=(11z)/(5)+(52)/(5)_27z=-79+25
Add 25 to -79 to get -54.
x=2y-4z-19_y=(11z)/(5)+(52)/(5)_27z=-54
Divide each term in the equation by 27.
x=2y-4z-19_y=(11z)/(5)+(52)/(5)_(27z)/(27)=-(54)/(27)
Simplify the left-hand side of the equation by canceling the common factors.
x=2y-4z-19_y=(11z)/(5)+(52)/(5)_z=-(54)/(27)
Simplify the right-hand side of the equation by simplifying each term.
x=2y-4z-19_y=(11z)/(5)+(52)/(5)_z=-2
Replace all occurrences of z with the solution found by solving the last equation for z. In this case, the value substituted is -2.
x=2y-4(-2)-19_y=(11z)/(5)+(52)/(5)_z=-2
Replace all occurrences of z with the solution found by solving the last equation for z. In this case, the value substituted is -2.
x=2y-4(-2)-19_y=(11(-2))/(5)+(52)/(5)_z=-2
Multiply -4 by each term inside the parentheses.
x=2y+8-19_y=(11(-2))/(5)+(52)/(5)_z=-2
Subtract 19 from 8 to get -11.
x=2y-11_y=(11(-2))/(5)+(52)/(5)_z=-2
Multiply 11 by -2 in the numerator.
x=2y-11_y=(11*-2)/(5)+(52)/(5)_z=-2
Multiply 11 by -2 to get -22.
x=2y-11_y=(-22)/(5)+(52)/(5)_z=-2
Move the minus sign from the numerator to the front of the expression.
x=2y-11_y=-(22)/(5)+(52)/(5)_z=-2
Complete the multiplication to produce a denominator of 5 in each expression.
x=2y-11_y=(52)/(5)-(22)/(5)_z=-2
Combine the numerators of all fractions that have common denominators.
x=2y-11_y=(52-22)/(5)_z=-2
Subtract 22 from 52 to get 30.
x=2y-11_y=(30)/(5)_z=-2
Reduce the expression (30)/(5) by removing a factor of 5 from the numerator and denominator.
x=2y-11_y=6_z=-2
Replace all occurrences of y with the solution found by solving the last equation for y. In this case, the value substituted is 6.
x=2(6)-11_y=6_z=-2
Multiply 2 by each term inside the parentheses.
x=12-11_y=6_z=-2
Subtract 11 from 12 to get 1.
x=1_y=6_z=-2
This is the solution to the system of equations.
x=1_y=6_z=-2
RELATED QUESTIONS
Solve the following system of equations using Gaussian elimination (with a matrix) and... (answered by checkley71,richwmiller)
Solve the equations by using elimination method and Use determinants to solve systems of... (answered by Theo)
Solve the system of linear equations using the Gauss-Jordan elimination method.
3x +... (answered by solver91311)
Solve the system of equations using either elimination or substitution
2x-3y+z=13... (answered by Alan3354)
How do you solve the system of three linear equations using the elimination method which... (answered by Edwin McCravy)
Use elimination method to solve the system.
-4x+4y-2z=-8
-3x-y+4z=0... (answered by ankor@dixie-net.com)
Can you help me solve the system of equations using matrices: x-2y+3z=7 2x+y+z=5... (answered by Jk22)
Solve the system of equations: {2x+y-3z=1}... (answered by jim_thompson5910)
Solve the linear system with a single solution or using the addtion and substitution... (answered by Alan3354)