Question

follow the step - by - step process to solve the linear equation.
a) determine the lcd of all denominators in the given linear equation.
b) write the new linear equation obtained after multiplying both sides of the original equation by the lcd.
c) solve the equation.
\\(3x+\frac{3}{2}=\frac{13}{5}\\)

Explanation:

Part (a)

Step1: Identify denominators

The denominators in the equation \(3x+\frac{3}{2}=\frac{13}{5}\) are \(2\) and \(5\).

Step2: Find LCD of 2 and 5

The least common denominator (LCD) of two numbers is the least common multiple (LCM) of those numbers. The LCM of \(2\) and \(5\) (since they are coprime) is \(2\times5 = 10\).

Step1: Multiply each term by LCD

Multiply each term of the equation \(3x+\frac{3}{2}=\frac{13}{5}\) by \(10\) (the LCD found in part (a)).
For the first term \(3x\): \(10\times3x=30x\)
For the second term \(\frac{3}{2}\): \(10\times\frac{3}{2}=\frac{30}{2} = 15\)
For the third term \(\frac{13}{5}\): \(10\times\frac{13}{5}=\frac{130}{5}=26\)

Step2: Write the new equation

Putting it all together, the new equation is \(30x + 15=26\).

Step1: Subtract 15 from both sides

Starting with \(30x+15 = 26\), subtract \(15\) from both sides: \(30x+15 - 15=26 - 15\)
Simplifying, we get \(30x=11\)

Step2: Divide by 30

Divide both sides by \(30\): \(x=\frac{11}{30}\)

Answer:

The LCD is \(10\).

Part (b)