Solve the system by substitution
-x-y-z=-8
-4z+4y+5z=7
2x+2z=4
well, it is clear that x = 8-y-z
plug that into the other two equations. Then make a final substitution there for y or z.
thank you
To solve the system using substitution, we'll solve one equation for one variable and substitute it into the other equations. Let's solve the second equation for y.
From equation 2: -4z + 4y + 5z = 7
Rearrange the terms: 4y = 7 + 4z
Divide both sides by 4: y = (7 + 4z)/4
Simplify: y = (7/4) + z
Now, substitute the expression for y into the other equations.
Substitute y = (7/4) + z into equation 1:
-x - [(7/4) + z] - z = -8
Simplify: -x - (7/4) - 2z = -8
Next, substitute y = (7/4) + z into equation 3:
2x + 2z = 4
Replace y with (7/4) + z: 2x + 2z = 4
Now we have a system of two equations with two variables:
1) -x - (7/4) - 2z = -8
2) 2x + 2z = 4
From equation 2, we can isolate x:
2x = 4 - 2z
Divide both sides by 2: x = 2 - z
Now, substitute the expression for x into equation 1:
- (2 - z) - (7/4) - 2z = -8
Simplify: -2 + z - 7/4 - 2z = -8
Rearrange the terms: -7/4 - z - 2z = -8 + 2
Combine like terms: -7/4 - 3z = -6
To solve for z, multiply through by 4 to eliminate the fraction:
-7 - 12z = -24
Add 7 to both sides: -12z = -17
Divide by -12: z = -17/-12 = 17/12
Now that we have z, we can substitute this value back into one of the equations to find the values of x and y.
Using equation 2:
2x + 2(17/12) = 4
Simplify: 2x + 17/6 = 4
Subtract 17/6 from both sides: 2x = 4 - 17/6
Combine like terms: 2x = 24/6 - 17/6 = 7/6
Divide by 2: x = 7/12
Finally, substitute values of x and z back into the expression for y:
y = (7/4) + z
y = (7/4) + (17/12)
Simplify: y = (21/12) + (17/12) = 38/12 = 19/6
Therefore, the solution to the system of equations is:
x = 7/12, y = 19/6, z = 17/12.