Which equation is an identity?
a. 8-(5x+2)=-5x-6
b. 7z+10-z=8z-2(z-5)
c. 8m-4=5m+8-m
d. 6y+5=6y-5
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To determine which equation is an identity, we need to find the equation where both sides are equal for all values of the variable(s). In other words, we are looking for an equation that is true regardless of the value(s) we substitute in.
Let's go through each option and simplify the equation to see if it qualifies as an identity:
a. 8 - (5x + 2) = -5x - 6
To simplify this equation, we distribute the negative sign inside the parentheses:
8 - 5x - 2 = -5x - 6
6 - 5x = -5x - 6
Here, we can see that the variable terms on both sides (-5x and -5x) cancel each other out.
6 = -6
This is a contradiction, which means that this equation is not an identity.
b. 7z + 10 - z = 8z - 2(z - 5)
To simplify this equation, we distribute the negative sign before the parentheses:
7z + 10 - z = 8z - 2z + 10
Combine like terms:
6z + 10 = 6z + 10
Here, we can see that both sides of the equation are equal, regardless of the value of z. This equation is an identity.
c. 8m - 4 = 5m + 8 - m
To simplify this equation, we combine the m terms:
8m - 4 = 4m + 8
Subtract 4m from both sides:
8m - 4m - 4 = 4m - 4m + 8
4m - 4 = 8
We can see here that this equation is not an identity since it depends on the value of m.
d. 6y + 5 = 6y - 5
To simplify this equation, we subtract 6y from both sides:
6y + 5 - 6y = 6y - 5 - 6y
5 = -5
This is a contradiction, and therefore, this equation is not an identity.
In conclusion, the equation b. 7z + 10 - z = 8z - 2(z - 5) is the only identity among the given options.