Solve the system of equations for each variable x, y, and z. Please show your work. (Hint: try the substitution method)
2x + y = 2
4x + z = 4y
y = x + 1
We are given the following system of equations:
1) 2x + y = 2
2) 4x + z = 4y
3) y = x + 1
We can use the substitution method to solve this system. We start by solving equation 3) for y:
3) y = x + 1
Next, we substitute equation 3) into equations 1) and 2):
1) 2x + (x + 1) = 2
2) 4x + z = 4(x + 1)
Simplifying equation 1):
2x + x + 1 = 2
3x + 1 = 2
3x = 1
x = 1/3
Substituting x = 1/3 into equation 3):
y = (1/3) + 1
y = 4/3
Substituting x = 1/3 into equation 2):
4(1/3) + z = 4[(1/3) + 1]
4/3 + z = 4/3 + 4
z = 4/3 + 4 - 4/3
z = 4
Therefore, the solution to the system of equations is:
x = 1/3
y = 4/3
z = 4
To solve this system of equations using the substitution method, we'll start by solving one equation for one variable and then substituting that into the other equations.
Let's start by solving the third equation, y = x + 1, for y:
y = x + 1
Now, we can substitute this expression for y into the other two equations.
Substituting y = x + 1 into the first equation, we have:
2x + (x + 1) = 2
3x + 1 = 2
3x = 2 - 1
3x = 1
x = 1/3
Now, substituting y = x + 1 into the second equation, we have:
4x + z = 4(y)
4(1/3) + z = 4(x + 1)
4/3 + z = 4/3 + 4
z = 4/3 + 4 - 4/3
z = 4
So, we have found x = 1/3 and z = 4.
Now, let's substitute these values back into one of the original equations to find y. Let's use the second equation:
4x + z = 4y
4(1/3) + 4 = 4y
4/3 + 4 = 4y
16/3 = 4y
y = (16/3) / 4
y = 4/3
Therefore, the solution to the system of equations is x = 1/3, y = 4/3, and z = 4.
To solve this system of equations using the substitution method, we can solve one equation for one variable and substitute it into the other equations.
Let's solve the third equation for y and express it in terms of x:
y = x + 1
Now we can substitute this expression for y into the first equation:
2x + (x + 1) = 2
Combining like terms:
3x + 1 = 2
Subtracting 1 from both sides:
3x = 1
Dividing both sides by 3:
x = 1/3
Now we substitute the value of x into the third equation to find y:
y = (1/3) + 1
y = 4/3
Now let's substitute the values of x and y into the second equation:
4(1/3) + z = 4(4/3)
Simplifying:
4/3 + z = 16/3
Subtracting 4/3 from both sides:
z = 16/3 - 4/3
Combining like terms:
z = 12/3
z = 4
So the solution to the system of equations is x = 1/3, y = 4/3, and z = 4.