Question

follow the step - by - step process to solve the linear equation.
a) determine the lcd of all denominators in the given linear equatio
b) write the new linear equation obtained after multiplying both sides
c) solve the equation.
\\(\frac{a - 2}{9}-\frac{5(a - 4)}{12}=\frac{5a + 4}{4}\\)

a) the lcd is \\(\square\\).
(simplify your answer.)

Explanation:

Step1: Find LCD of 9,12,4

Prime factors: $9=3^2$, $12=2^2\times3$, $4=2^2$. LCD is $2^2\times3^2=36$.

Step2: Multiply all terms by 36

$$36\times\frac{a-2}{9} - 36\times\frac{5(a-4)}{12} = 36\times\frac{5a+4}{4}$$
Simplify each term: $4(a-2) - 3\times5(a-4) = 9(5a+4)$

Step3: Expand all parentheses

$$4a - 8 -15a +60 = 45a +36$$

Step4: Combine like terms left side

$$-11a +52 = 45a +36$$

Step5: Isolate variable terms

$$52 -36 = 45a +11a$$
$$16 = 56a$$

Step6: Solve for $a$

$$a=\frac{16}{56}=\frac{2}{7}$$

Answer:

a) The LCD is $\boldsymbol{36}$.
b) The new equation is $\boldsymbol{4(a-2) - 15(a-4) = 9(5a+4)}$ (or expanded form $\boldsymbol{-11a + 52 = 45a + 36}$).
c) $\boldsymbol{a=\frac{2}{7}}$