Solve the system by elimination:
-2x + 2y + 3z = 0
-2x - y + z =-3
2x + 3y + 3z = 5
How do I do this? I need steps to understand please....
Well, let's tackle this system together! We'll eliminate one variable at a time until we have a solution.
Step 1: Choose one equation to start with. Let's take the first equation: -2x + 2y + 3z = 0.
Step 2: Now, compare this equation with the second and third equations.
- Compare the first equation with the second equation: -2x + 2y + 3z = 0 and -2x - y + z = -3.
Since the coefficients of x in both equations are the same, we can directly subtract the second equation from the first one:
(-2x + 2y + 3z) - (-2x - y + z) = 0 - (-3).
Simplify this equation: -2x + 2y + 3z + 2x + y - z = 0 + 3.
Combining like terms, we have: 2y + 2z = 3.
- Compare the first equation with the third equation: -2x + 2y + 3z = 0 and 2x + 3y + 3z = 5.
Since the coefficients of x in both equations are the same, we can directly subtract the third equation from the first one:
(-2x + 2y + 3z) - (2x + 3y + 3z) = 0 - 5.
Simplify this equation: -2x + 2y + 3z - 2x - 3y - 3z = 0 - 5.
Combining like terms, we have: -y = -5.
Step 3: Now we have two new equations:
- 2y + 2z = 3
- y = -5
Step 4: We can substitute the value of y from the second equation into the first equation to eliminate y:
2(-5) + 2z = 3.
Simplifying, we get: -10 + 2z = 3.
Adding 10 to both sides, we have: 2z = 13.
Finally, divide both sides by 2: z = 6.5.
Step 5: Substituting the value of z back into the second equation, we find the value of y:
y = -5.
Step 6: Now, substitute the values of y and z into any of the original equations to find the value of x. Let's use the first equation:
-2x + 2(-5) + 3(6.5) = 0.
Simplifying, we get: -2x - 10 + 19.5 = 0.
Combining like terms, we have: -2x + 9.5 = 0.
Subtracting 9.5 from both sides, we get: -2x = -9.5.
Finally, divide both sides by -2: x = 4.75.
So, the solution to the system of equations is:
x = 4.75
y = -5
z = 6.5
I hope that helps! Let me know if there's anything else I can assist you with.
To solve a system of equations by elimination, follow these steps:
Step 1: Rearrange the equations
Arrange the equations in a way that the coefficients of the variables line up vertically.
The given system of equations is:
-2x + 2y + 3z = 0 --> Equation A
-2x - y + z = -3 --> Equation B
2x + 3y + 3z = 5 --> Equation C
Step 2: Choose two equations to eliminate one variable
Select two equations and one variable to eliminate. Look for coefficients that, when multiplied, result in equal magnitude but opposite signs.
Let's eliminate x by multiplying Equation A by -1 and Equation B by -1:
-(-2x + 2y + 3z) = -(0)
-(-2x - y + z) = -(-3)
This gives us:
2x - 2y - 3z = 0 --> Equation D
2x + y - z = 3 --> Equation E
Step 3: Add or subtract the equations to eliminate the chosen variable
By adding Equation D and Equation E, the x variable will be eliminated:
(2x - 2y - 3z) + (2x + y - z) = 0 + 3
Simplifying the equation:
4x - y - 4z = 3 --> Equation F
Step 4: Repeat steps 2 and 3 with different variables until only one variable remains
Now, let's eliminate y by multiplying Equation E by -4:
-4(2x + y - z) = -4(3)
This gives us:
-8x - 4y + 4z = -12 --> Equation G
By adding Equation F and Equation G, the y variable will be eliminated:
(4x - y - 4z) + (-8x - 4y + 4z) = 3 + (-12)
Simplifying the equation:
-4x - 5y = -9 --> Equation H
Step 5: Solve the remaining equation for the last variable
Now, solve Equation H for y:
-4x - 5y = -9
Rearrange the equation:
-5y = 9 - 4x
Divide both sides of the equation by -5:
y = -(9 - 4x) / 5
Simplify:
y = (4x - 9) / 5
Step 6: Substitute the solved variable back into one of the original equations
Substitute y = (4x - 9) / 5 into Equation B:
-2x - (4x - 9) / 5 + z = -3
Multiply every term by 5 to eliminate the fraction:
-10x - 4x + 9 + 5z = -15
Simplifying the equation:
-14x + 5z = -24 --> Equation I
Step 7: Solve the remaining equation for the last variable
Now, solve Equation I for z:
-14x + 5z = -24
Rearrange the equation:
5z = 24 + 14x
Divide both sides of the equation by 5:
z = (24 + 14x) / 5
Simplify:
z = (12 + 7x) / 5
Step 8: Substitute the values of y and z back into one of the original equations
Substitute y = (4x - 9) / 5 and z = (12 + 7x) / 5 into Equation A:
-2x + 2((4x - 9) / 5) + 3((12 + 7x) / 5) = 0
Multiply every term by 5 to eliminate the fraction:
-10x + 2(4x - 9) + 3(12 + 7x) = 0
Simplifying the equation:
-10x + 8x - 18 + 36 + 21x = 0
Combine like terms:
19x + 18 = 0
Subtract 18 from both sides of the equation:
19x = -18
Divide both sides of the equation by 19 to solve for x:
x = -18 / 19
Step 9: Substitute the value of x back into the expressions for y and z to find their values
Substitute x = -18 / 19 into y = (4x - 9) / 5:
y = (4(-18 / 19) - 9) / 5
Simplify:
y = (-72 / 19 - 9) / 5
Find a common denominator:
y = (-72 / 19 - 9 * 19 / 19) / 5
Simplifying the expression:
y = (-72 - 171) / 95
y = -243 / 95
Substitute x = -18 / 19 into z = (12 + 7x) / 5:
z = (12 + 7(-18 / 19)) / 5
Simplify:
z = (12 - 126 / 19) / 5
Find a common denominator:
z = (12 * 19 / 19 - 126 / 19) / 5
Simplifying the expression:
z = (228 - 126) / 95
z = 102 / 95
Therefore, the solution to the given system of equations is:
x = -18 / 19
y = -243 / 95
z = 102 / 95
To solve the system by elimination, you need to eliminate one variable at a time by adding or subtracting the equations.
Here are the steps to solve the given system of equations:
Step 1: Rearrange the equations to align the variables in columns and constants in a separate column:
Equation 1: -2x + 2y + 3z = 0
Equation 2: -2x - y + z = -3
Equation 3: 2x + 3y + 3z = 5
Step 2: Choose two equations and eliminate one variable by adding or subtracting them. Let's eliminate the variable 'x' from equations 1 and 2.
Add Equation 1 and Equation 2:
(-2x + 2y + 3z) + (-2x - y + z) = 0 + (-3)
-4x + y + 4z = -3
Step 3: Choose different equations and eliminate the same variable again. Let's eliminate the variable 'x' from equations 2 and 3.
Multiply Equation 2 by 2 and Equation 3 by -2 to make the coefficients of 'x' cancel each other out:
2 * (-2x - y + z) = 2 * (-3)
-4x - 2y + 2z = -6
-2 * (2x + 3y + 3z) = -2 * 5
-4x - 6y - 6z = -10
Add the above two equations:
(-4x - 2y + 2z) + (-4x - 6y - 6z) = -6 + (-10)
-8x - 8y - 4z = -16
Step 4: You now have two equations without the variable 'x':
-4x + y + 4z = -3
-8x - 8y - 4z = -16
Step 5: Continue eliminating variables by either adding or subtracting equations. Since we have a -4z term present in one equation and a +4z term in the other equation, we can eliminate the 'z' variable.
Multiply Equation 1 by 2 and Equation 2 by 1 to make the coefficients of 'z' cancel each other out:
2 * (-4x + y + 4z) = 2 * (-3)
-8x + 2y + 8z = -6
1 * (-8x - 8y - 4z) = 1 * (-16)
-8x - 8y - 4z = -16
Add the above two equations:
(-8x + 2y + 8z) + (-8x - 8y - 4z) = -6 + (-16)
-16x - 6y + 4z = -22
Step 6: You now have two equations without the variables 'x' and 'z':
-16x - 6y + 4z = -22
-8x - 8y - 4z = -16
Step 7: Choose the remaining two equations and eliminate one more variable. Let's eliminate the variable 'y' from the above equations.
Multiply Equation 1 by -1 and Equation 2 by -2 to make the coefficients of 'y' cancel each other out:
-1 * (-16x - 6y + 4z) = -1 * (-22)
16x + 6y - 4z = 22
-2 * (-8x - 8y - 4z) = -2 * (-16)
16x + 16y + 8z = 32
Add the above two equations:
(16x + 6y - 4z) + (16x + 16y + 8z) = 22 + 32
32x + 22z = 54
Step 8: You now have an equation with only the variables 'x' and 'z':
32x + 22z = 54
Step 9: Solve the equation obtained in step 8 to find the values of 'x' and 'z'.
Step 10: Once you find the values of 'x' and 'z', substitute them back into any of the original equations to solve for the remaining variable 'y'.
By following these steps, you can solve the given system of equations by elimination.
(1a) -2x + 2y + 3z = 0
(2a) -2x - y + z =-3
(3a) 2x + 3y + 3z = 5
2a from 1a and add 1a to 3a to get
(1b) 3y + 2z = 3
(2b) 5y + 6z = 5
Now, just by inspection, we see that y=1,z=0 fit these two equations. Or, you can subtract 2b from 3 times 1b to eliminate z.
In any case, just use any of the 1st 3 equations, knowing y=1 z=0, to get x.