5y-4 upon 6 +3y+10 upon 2 =4y+1 solve the following Linea equation using transposition method

Let's solve the given linear equation step by step using the transposition method.

The equation is:

\[
\frac{5y - 4}{6} + \frac{3y + 10}{2} = 4y + 1
\]

### Step 1: Eliminate the fractions
To eliminate the fractions, we need to find the least common multiple (LCM) of the denominators, which are 6 and 2. The LCM of 6 and 2 is 6.

Multiply the entire equation by 6 to eliminate the denominators:

\[
6 \times \left(\frac{5y - 4}{6}\right) + 6 \times \left(\frac{3y + 10}{2}\right) = 6 \times (4y + 1)
\]

This simplifies to:

\[
(5y - 4) + 3 \times (3y + 10) = 6 \times 4y + 6 \times 1
\]

### Step 2: Simplify the equation
Now, distribute the 3 on the left-hand side and simplify:

\[
5y - 4 + 9y + 30 = 24y + 6
\]

Combine like terms:

\[
14y + 26 = 24y + 6
\]

### Step 3: Transpose the terms
Now, we'll transpose the terms to isolate \(y\) on one side of the equation. First, move the \(24y\) term to the left side by subtracting \(24y\) from both sides:

\[
14y - 24y + 26 = 6
\]

This simplifies to:

\[
-10y + 26 = 6
\]

Next, move the constant term \(26\) to the right side by subtracting \(26\) from both sides:

\[
-10y = 6 - 26
\]

Simplifying further:

\[
-10y = -20
\]

### Step 4: Solve for \(y\)
Finally, divide both sides by \(-10\) to solve for \(y\):

\[
y = \frac{-20}{-10} = 2
\]

### Final Answer:
The solution to the equation is \(y = 2\).

Would you like any additional details or have any questions?

Here are some related questions:
1. How do you solve linear equations with variables on both sides?
2. What are the different methods to solve linear equations besides transposition?
3. Can you explain how to handle equations with multiple fractions?
4. What are the steps to check the solution of a linear equation?
5. How do you graphically represent the solution of a linear equation?

**Tip:** Always check your solution by substituting it back into the original equation to verify it satisfies the equation.

Learn how to solve a linear equation involving fractions and using the transposition method step-by-step. This problem covers key algebraic concepts, including linear equations, fraction operations, and the transposition method. Suitable for students in Grades 8-10.

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