z = x-y+4
z = y-w-3
z = w-x+5
Based on the system of equations above, what is the value of z?
a) 2
b) 3
c) 4
d) 6
e) 12
add up all 3 equations ....
3z = x-y+4 + y-w-3 + w-x+5
3z = 6
z = 2
To find the value of z, we need to solve the system of equations.
Given:
z = x - y + 4 ......(Equation 1)
z = y - w - 3 ......(Equation 2)
z = w - x + 5 ......(Equation 3)
Let's solve this system of equations step by step:
Step 1: Let's start with Equation 1 and Equation 2.
Equation 1: z = x - y + 4
Equation 2: z = y - w - 3
Since both equations are equal to z, we can equate them:
x - y + 4 = y - w - 3
Step 2: Now, let's rearrange the equation to isolate x and w.
x = y - w - 3 - 4
x = y - w - 7 .........(Equation 4)
Step 3: Now, let's substitute Equation 4 into Equation 3.
z = w - x + 5
z = w - (y - w - 7) + 5
z = w - y + w + 7 + 5
z = 2w - y + 12 .........(Equation 5)
Step 4: Now, let's substitute Equation 2 into Equation 5.
z = 2w - y + 12
z = 2(y - w - 3) - y + 12
z = 2y - 2w - 6 - y + 12
z = y - 2w + 6 .........(Equation 6)
Step 5: Now, let's equate Equation 5 and Equation 6.
2w - y + 12 = y - 2w + 6
Step 6: Now, let's rearrange the equation to isolate w.
2w + 2w = y + y + 6 - 12
4w = 2y - 6
2w = y - 3
Step 7: Now, let's substitute the value of w into Equation 6.
z = y - 2(y - 3) + 6
z = y - 2y + 6 + 6
z = -y + 12
Step 8: Now, let's equate the value of z from Equation 1.
z = x - y + 4
-y + 12 = x - y + 4
Step 9: Now, let's rearrange the equation to isolate x.
x = -y + 12 + y - 4
x = 8
Step 10: Finally, let's substitute the value of x into Equation 1 to find the value of z.
z = x - y + 4
z = 8 - y + 4
z = 12 - y
Since z = 12 - y, we cannot determine the exact value of z without knowing the value of y. Therefore, the answer is not provided in the given options.
To find the value of z, we need to solve the system of equations given. Let's go step by step:
Step 1: Start by isolating one variable in terms of the others.
From the first equation, we can isolate x:
x = z + y - 4
Step 2: Substitute the isolated variable into the other equations.
Let's substitute x into the second equation:
z = y - w - 3
=> z = (z + y - 4) - w - 3
=> z = z + y - w - 7
Step 3: Simplify the equation.
Rearrange the equation to isolate z:
z - z = y - w - 7
0 = y - w - 7
y - w = 7
Step 4: Substitute the simplified equation into the third equation.
Let's substitute y - w = 7 into the third equation:
z = w - (z + y - 4) + 5
=> z = w - z - y + 4 + 5
=> z = -z - y + w + 9
Step 5: Rearrange the equation to isolate z.
z + z = -y + w + 9
2z = -y + w + 9
Now we have two equations:
y - w = 7 (Equation 1)
2z = -y + w + 9 (Equation 2)
Step 6: Solve the system of equations.
We can use either substitution or elimination method to solve the system of equations. Let's use substitution:
From Equation 1, we can express y in terms of w:
y = 7 + w
Substitute y into Equation 2:
2z = -(7 + w) + w + 9
2z = -7 - w + w + 9
2z = 2
Divide both sides by 2:
z = 2/2
z = 1
Therefore, the value of z is 1.