Question

Solve the equation for q. State whether the equation is a​ contradiction, an​ identity, or a conditional equation.
StartFraction q Over 3 EndFraction plus three fourths equals StartFraction 4 q plus 9 Over 12 EndFraction
q
3+
3
4=
4q+9
12

Answer 1

To solve the equation

\[
\frac{q}{3} + \frac{3}{4} = \frac{4q + 9}{12},
\]

first, we will eliminate the fractions by finding the least common multiple (LCM) of the denominators (3, 4, and 12). The LCM is 12.

Now, we will multiply every term in the equation by 12:

\[
12 \cdot \left( \frac{q}{3} \right) + 12 \cdot \left( \frac{3}{4} \right) = 12 \cdot \left( \frac{4q + 9}{12} \right).
\]

This simplifies to:

\[
4q + 9 = 4q + 9.
\]

Next, let's simplify the equation:

Subtract \(4q\) from both sides:

\[
4q + 9 - 4q = 4q + 9 - 4q,
\]
which simplifies to:

\[
9 = 9.
\]

Since this statement is always true, it means that the original equation is true for all values of \(q\). Therefore, this equation is an **identity**.

In conclusion:
- The solution: **All values of** \(q\)
- The type of equation: **Identity**.