Favorite Math Problem Solution
Equations:
1. x + y + z = 6
2. x + 2y = 5
3. 2x + z = 5
Step 1: Eliminate x from equations 2 and 3. Multiply equation 2 by -1 and add it to equation 1 to eliminate x:
-x - 2y = -5 (equation 2 multiplied by -1)
x + y + z = 6 (equation 1)
-y + z = 1 (resultant equation)
Multiply equation 3 by -1 and add it to equation 1 to eliminate x:
-2x - z = -5 (equation 3 multiplied by -1)
x + y + z = 6 (equation 1)
-y + 2z = 1 (resultant equation)
Step 2: Eliminate y from equations 1 and the resultant equation from step 1.
Multiply equation 1 by -1 and add it to the resultant equation from step 1 to eliminate y:
-x - y - z = -6 (equation 1 multiplied by -1)
-y + z = 1 (resultant equation from step 1)
-x - 2y = -5 (resultant equation)
Multiply equation 1 by -1 and add it to the resultant equation from step 2 to eliminate y: -x - y - z = -6 (equation 1 multiplied by -1) -y + 2z = 1 (resultant equation from step 2)
-x + z = -5 (resultant equation)
Step 3: Solve the resultant equations for x and z.
-y + z = 1 (resultant equation from step 1)
-x + z = -5 (resultant equation from step 2)
Adding the above two equations, we get:
-x - y + 2z = -4 (equation 7)
Substituting x = -5 + z into equation 7, we get:
-(-5 + z) - y + 2z = -4
Simplifying, we get:
5 - z - y + 2z = -4
3z - y = -9 (equation 8)
Step 4: Substitute the value of z from equation 8 into equation 1 to find the value of y.
-y + z = 1 (resultant equation from step 1)
3z - y = -9 (equation 8)
Substituting z = (1 + y)/2 from equation 1 into equation 8, we get:
3((1 + y)/2) - y = -9
Simplifying, we get:
3 + 3y - 2y = -18
y = -21
Step 5: Substitute the values of y and z from equations 1 and 8 into equation 2 to find the value of x.
x + 2y = 5 (equation 2)
y = -21 (value of y from step 4)
z = 1 (value of z from equation 1)
Substituting y = -21 and z = 1 into equation 2, we get:
x + 2(-21) = 5
x - 42 = 5
x =47
Equations:
1. x + y + z = 6
2. x + 2y = 5
3. 2x + z = 5
Step 1: Eliminate x from equations 2 and 3. Multiply equation 2 by -1 and add it to equation 1 to eliminate x:
-x - 2y = -5 (equation 2 multiplied by -1)
x + y + z = 6 (equation 1)
-y + z = 1 (resultant equation)
Multiply equation 3 by -1 and add it to equation 1 to eliminate x:
-2x - z = -5 (equation 3 multiplied by -1)
x + y + z = 6 (equation 1)
-y + 2z = 1 (resultant equation)
Step 2: Eliminate y from equations 1 and the resultant equation from step 1.
Multiply equation 1 by -1 and add it to the resultant equation from step 1 to eliminate y:
-x - y - z = -6 (equation 1 multiplied by -1)
-y + z = 1 (resultant equation from step 1)
-x - 2y = -5 (resultant equation)
Multiply equation 1 by -1 and add it to the resultant equation from step 2 to eliminate y: -x - y - z = -6 (equation 1 multiplied by -1) -y + 2z = 1 (resultant equation from step 2)
-x + z = -5 (resultant equation)
Step 3: Solve the resultant equations for x and z.
-y + z = 1 (resultant equation from step 1)
-x + z = -5 (resultant equation from step 2)
Adding the above two equations, we get:
-x - y + 2z = -4 (equation 7)
Substituting x = -5 + z into equation 7, we get:
-(-5 + z) - y + 2z = -4
Simplifying, we get:
5 - z - y + 2z = -4
3z - y = -9 (equation 8)
Step 4: Substitute the value of z from equation 8 into equation 1 to find the value of y.
-y + z = 1 (resultant equation from step 1)
3z - y = -9 (equation 8)
Substituting z = (1 + y)/2 from equation 1 into equation 8, we get:
3((1 + y)/2) - y = -9
Simplifying, we get:
3 + 3y - 2y = -18
y = -21
Step 5: Substitute the values of y and z from equations 1 and 8 into equation 2 to find the value of x.
x + 2y = 5 (equation 2)
y = -21 (value of y from step 4)
z = 1 (value of z from equation 1)
Substituting y = -21 and z = 1 into equation 2, we get:
x + 2(-21) = 5
x - 42 = 5
x =47