solve by elimination
5x+3y+5z=-16
2x+5y-z=-26
x-4y+2z=10
1st - 5(3rd) ---> 23y - 5z = -66
2nd - 2(3rd) ---> 13y - 5z = -36
subtract those ----> 10y = 30
y = 3
back in 13y-5z=-36
39 - 5y = -36
-5z = 75
z = -15
put y=3 and z=-15 into the original 3rd and you are done.
To solve this system of equations by elimination, follow these steps:
Step 1: Choose two equations and select variables to eliminate.
In this case, we'll choose the first and second equations and eliminate the variable x.
Step 2: Multiply the equations by appropriate coefficients to make the coefficients of one of the variables opposite.
Multiply the second equation by 5 and the first equation by 2:
10x + 25y - 5z = -130 (equation 1 multiplied by 2)
10x + 30y + 50z = -80 (equation 2 multiplied by 5)
Step 3: Subtract one equation from the other to eliminate the variable.
Subtract equation 2 from equation 1:
(10x + 25y - 5z) - (10x + 30y + 50z) = -130 - (-80)
Simplifying, we get:
-5y - 55z = -50
Step 4: Now, choose two different equations and eliminate another variable.
We'll choose equations 2 and 3 and eliminate the variable x.
Multiply the first equation by 2 and the third equation by 5:
10x + 25y - 5z = -130 (equation 1 multiplied by 2)
5x - 20y + 10z = 50 (equation 3 multiplied by 5)
Step 5: Subtract one equation from the other to eliminate the variable.
Subtract equation 3 from equation 1:
(10x + 25y - 5z) - (5x - 20y + 10z) = -130 - 50
Simplifying, we get:
5x + 45y - 15z = -180
Step 6: Now, we have two equations with two variables:
-5y - 55z = -50 (from step 3)
5x + 45y - 15z = -180 (from step 5)
Step 7: Solve this new system of equations using any method you prefer, such as substitution or elimination.
To use elimination, we'll multiply the first equation by -1 and add it to the second equation:
5x + 45y - 15z + (5y + 55z) = -180 + 50
5x + 50y + 40z = -130
Now, we have only two variables left to solve for: y and z.
Step 8: Solve the new equation for y.
5x + 50y + 40z = -130
50y = -5x - 40z - 130
y = (-5x - 40z - 130)/50
y = -x/10 - 4z/10 - 13/5
Step 9: Substitute the expression for y back into one of the original equations to solve for z.
Using the first equation: 5x + 3y + 5z = -16
5x + 3(-x/10 - 4z/10 - 13/5) + 5z = -16
5x - 3x/10 - 12z/10 - 39/5 + 5z = -16
(50x - 3x - 12z + 50z)/10 - 39/5 = -16
(47x + 38z - 39)/10 = -16
47x + 38z - 39 = -160
47x + 38z = -121
Step 10: Solve the obtained equation for z.
47x + 38z = -121
z = (-121 - 47x)/38
Step 11: Substitute the value of z into the expression for y obtained in Step 8 to find y.
y = -x/10 - 4z/10 - 13/5
y = -x/10 - 4(-121 - 47x)/10*38 - 13/5
y = -x/10 + 2(-121 - 47x)/19*5 - 13/5
Step 12: Choose any value for x and substitute it into the expressions for y and z to get the complete solution.
For example, let x = 0:
y = -0/10 + 2(-121 - 47*0)/19*5 - 13/5 = -2.7
z = (-121 - 47*0)/38 = -3.184
The complete solution is x = 0, y = -2.7, z = -3.184.