Final answer:
The equation 9(x - 4) = 9x - 36 is an identity, as both sides reduce to the same expression, indicating it has infinitely many solutions. This is because simplifying both sides of the equation yields a true statement, 0 = 0.
Step-by-step explanation:
To solve the equation 9(x - 4) = 9x - 36, we can distribute the 9 on the left side and then simplify both sides:
9x - 36 = 9x - 36
After simplification, we see that both sides of the equation are identical, which gives us 0 = 0. This means that the original equation is true for all values of x. Therefore, the equation represents an identity, and there are infinitely many solutions.
The significance of the equation representing an identity
An equation that simplifies to a true statement, such as 0 = 0, indicates that the original equation is valid for any real number substituted in place of the variable. This means we have an identity equation that reflects a fundamental truth in mathematics rather than a conditional equation that's only true under specific circumstances.
This concept is parallel to constructing a table or graph for a specific line equation like y = 9 + 3x, where we substitute different x values to find corresponding y values. However, in the case of an identity, every value of x satisfies the equation, unlike a linear equation where each x pairs with a unique y.