PLEASE SHOW ALL THE STEPS THAT YOU USE TO SOLVE THIS PROBLEM
solve the system by elimination
-2x+2y+3z=0
-2x-y+z=-3
2x+3y+3z=5
Subtract Eq2 from Eq1:
Diff. = 3y + 2z = 3.
Add Eq2 and Eq3:
Sum = 2y + 4z = 2.
3y + 2z = 3. Multiply by 2 and subtract.
2y + 4z = 2.
6y + 4z = 6.
2y + 4z = 2.
Diff. : 4y = 4.
Y = 1.
6y + 4z = 6,
6*1 + 4z = 6,
Z = 0.
Eq1: -2x + 2y + 3z = 0.
-2x + 2*1 + 3*0 = 0,
X = 1.
Solution = (x, y, z) = (1, 1, 0).
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To solve this system of equations by elimination, we'll eliminate one variable at a time by adding or subtracting the equations. Here are the steps to solve it:
Step 1: Write down the system of equations:
-2x + 2y + 3z = 0 ...(equation 1)
-2x - y + z = -3 ...(equation 2)
2x + 3y + 3z = 5 ...(equation 3)
Step 2: Begin by eliminating the x-variable:
- Multiply equation 2 by -1 to change the signs:
2x + y - z = 3 ...(equation 2 multiplied by -1)
- Now, add equation 1 and equation 2 together:
(-2x + 2y + 3z) + (2x + y - z) = 0 + 3
Simplify the equation:
3y + 2z = 3 ...(equation 4)
Step 3: Eliminate another variable (preferably z in this case):
- Multiply equation 1 by 3 and equation 3 by 2 in order to cancel out z when added:
(-2x + 2y + 3z) * 3 + (2x + 3y + 3z) * 2 = 0 * 3 + 5 * 2
Simplify the equation:
-6x + 6y + 9z + 4x + 6y + 6z = 0 + 10
Combine like terms:
-2x + 12y + 15z = 10 ...(equation 5)
Step 4: Now we have two new equations (equations 4 and 5) that can be solved simultaneously.
Simplify equations 4 and 5:
3y + 2z = 3 ...(equation 4)
-2x + 12y + 15z = 10 ...(equation 5)
Step 5: Eliminate y-variable by multiplying equation 4 by -4:
-4(3y + 2z) = -4(3)
-12y - 8z = -12 ...(equation 6)
Step 6: Add equation 5 and equation 6:
(-2x + 12y + 15z) + (-12y - 8z) = 10 + (-12)
Simplify the equation:
-2x + 7z = -2 ...(equation 7)
Step 7: Now we have two new equations (equations 6 and 7) that can be solved simultaneously.
Simplify equations 6 and 7:
-12y - 8z = -12 ...(equation 6)
-2x + 7z = -2 ...(equation 7)
Step 8: Solve the system of equations given in equations 6 and 7. You can use various methods like substitution or elimination to find the values of y and z.
By following these steps, you can solve the system of equations using elimination.
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