Final answer:
To solve the system of equations x + 9y + z = 20, x + 10y – 2z = 18, and 3x + 27y + 2z = 58, we can use the substitution method. First, we can solve for x in terms of y and z in one of the equations and substitute it into the other equations. By simplifying and solving the resulting equations, we find that the values of x, y, and z are -18, 4, and 2, respectively.
Step-by-step explanation:
To solve the system of equations:
x + 9y + z = 20
x + 10y – 2z = 18
3x + 27y + 2z = 58
We can use the method of elimination or substitution. Let's use the substitution method:
From the first equation, we can rewrite it as x = 20 - 9y - z.
Substituting this value of x into the second equation, we get:
(20 - 9y - z) + 10y - 2z = 18.
Simplifying this equation, we get 10 - y - 3z = 18.
From the third equation, we can rewrite it as x = (58 - 27y - 2z) / 3.
Substituting this value of x into the first equation, we get:
(58 - 27y - 2z) / 3 + 9y + z = 20.
Simplifying this equation, we get 58 - 27y - 2z + 27y + 3z = 60.
Combining like terms, we get z = 2.
Substituting this value of z into the second equation, we get:
x + 10y - 2(2) = 18.
Simplifying this equation, we get x + 10y - 4 = 18.
Combining like terms, we get x + 10y = 22.
Substituting the value of x from the first equation, we get:
(20 - 9y - 2) + 10y = 22.
Simplifying this equation, we get 20 - 9y - 2 + 10y = 22.
Combining like terms, we get y = 4.
Finally, substituting the values of x and y into the first equation, we get:
20 - 9(4) - 2 = 20 - 36 - 2 = -18.
Therefore, the solution to the system of equations is x = -18, y = 4, and z = 2.