Solve by elimination
-2x+2y+3z=0
-2x-y+z=-3
2x+3y+3z=5
Please help. I have 5 pages and 3 hours spent and found the question on Google, but the calculations did not add up correctly.
Thank you
Step 1: Multiply first equation by −1 and add the result to the second equation. The result is:
-2x+2y+3z=0
-3y-2z=-3
2x+3y+3z=5
Step 2: Multiply first equation by 1 and add the result to the third equation. The result is:
-2x+2y+3z=0
-3y-2z=-3
5y+6z=5
Step 3: Multiply second equation by 3 and add the result to the third equation. The result is:
-2x+2y+3z=0
-3y-2z=-3
-4y=-4
Step 4: solve for y.
-4y=-4
y=1
Step 5: plug y in and solve for z.
-3y-2z=-3
-2z=0
z=0
Step 6: solve for x by substituting y=1 and z=0 into the first equation.
Multiply first equation by − 1
The result: 2 x - 2 y - 3 z = 0
Add the result to the second equation
2 x - 2 y - 3 z = 0
+
- 2 x - y + z = - 3
_______________
0 - 3 y - 2 z = - 3
- 3 y - 2 z = - 3
Add the first equation to the third equation
- 2 x + 2 y + 3 z = 0
+
2 x + 3 y + 3 z = 5
_________________
0 + 5 y + 6 z = 5
5 y + 6 z = 5
Now you have system of two equatios:
- 3 y - 2 z = - 3
5 y + 6 z = 5
- 3 y - 2 z = - 3 Multiply both sides by 3
- 9 y - 6 z = - 9
- 9 y - 6 z = - 9
+
5 y + 6 z = 5
____________
4 y + 0 = 4
4 y = 4 Divide both sides by 4
y = 4 / 4
y = 1
Replace y = 1 in equation:
- 3 y - 2 z = - 3
- 3 * 1 - 2 z = 3
- 3 - 2 z = - 3 Add 3 to both sides
- 3 - 2 z + 3 = - 3 + 3
2 z = 0 Divide both sides by 2
z = 0
Replace y = 1 and z = 0 in equation:
2 x + 3 y + 3 z = 5
2 x + 3 * 1 + 3 * 0 = 5
2 x + 3 = 5 Subtract 3 to both sides
2 x + 3 - 3 = 5 - 3
2 x = 2 Divide both sides by 2
x = 2 / 2
x = 1
The solutions are x = 1 , y = 1 , z = 0
or, for a more condensed presentation, type in your coefficients here and see all the details.
http://www.gregthatcher.com/Mathematics/GaussJordan.aspx
To solve this system of equations using the elimination method, you need to eliminate one variable at a time until you can solve for the remaining variables.
Step 1: Choose two equations and eliminate one variable.
Let's eliminate the variable x by adding the first equation and the second equation together:
(-2x + 2y + 3z) + (-2x - y + z) = 0 + (-3)
Simplifying this equation, we get:
-4x + y + 4z = -3 ........... (Equation 3)
Step 2: Choose two other equations and eliminate the same variable.
Let's eliminate the variable x by adding the first equation and the third equation together:
(-2x + 2y + 3z) + (2x + 3y + 3z) = 0 + 5
Simplifying this equation, we get:
5y + 6z = 5 ........... (Equation 4)
Now we have two equations (Equation 3 and Equation 4) with two variables (y and z), so we will solve these equations simultaneously.
Step 3: Solve the system of equations formed in Step 2.
Multiply Equation 3 by 5 to make the coefficients of y in Equation 3 and Equation 4 equal:
-20x + 5y + 20z = -15 ........... (Equation 5)
Now, subtract Equation 4 from Equation 5:
-20x + 5y + 20z - (5y + 6z) = -15 - 5
Simplifying this equation, we get:
-20x + 14z = -20 ........... (Equation 6)
Step 4: Solve Equation 6 for z.
To do this, we isolate z by getting it on one side of the equation:
-20x + 14z = -20
14z = 20x - 20
Divide both sides by 14:
z = (20x - 20) / 14
Simplifying this equation, we get:
z = (10x - 10) / 7 ........... (Equation 7)
Step 5: Substitute the expression for z from Equation 7 into Equation 4 and solve for y.
5y + 6z = 5
5y + 6((10x - 10) / 7) = 5
Multiply both sides by 7 to eliminate the denominator:
35y + 60x - 60 = 35
35y + 60x = 95
Subtract 60x from both sides:
35y = -60x + 95
Divide both sides by 35:
y = (-60x + 95) / 35
Simplifying this equation, we get:
y = (-12x + 19) / 7 ........... (Equation 8)
So now we have expressions for y and z in terms of x.
Step 6: Substitute the expressions for y and z from Equations 7 and 8 into any of the original equations.
Let's substitute them into the first equation:
-2x + 2y + 3z = 0
-2x + 2((-12x + 19) / 7) + 3((10x - 10) / 7) = 0
Multiply through by 7 to eliminate the denominator:
-14x + 14((-12x + 19) / 7) + 21((10x - 10) / 7) = 0
Simplify the equation:
-14x - 24x + 38 + 30x - 30 = 0
Combine like terms:
-8x + 8 = 0
Add 8x to both sides:
8 = 8x
Divide by 8:
x = 1
Now that we have the value of x, we can substitute it back into Equations 7 and 8 to find the values of y and z.
Substituting x = 1 into Equation 7:
z = (10(1) - 10) / 7
z = 0
Substituting x = 1 into Equation 8:
y = (-12(1) + 19) / 7
y = 1
So the solution to the system of equations is x = 1, y = 1, and z = 0.
I apologize if there were any mistakes or if the previous calculations did not add up correctly. It's always a good idea to double-check your work to ensure accuracy.