Let's solve the equation step by step:

$\frac{3(7m - 1)}{4} - \frac{2m - 1 - m}{2} = m + \frac{3}{2}$

Step 1: Simplify both sides

Left-hand side (LHS):

We begin by simplifying the terms on the left-hand side of the equation.

1. Expand $\frac{3(7m - 1)}{4}$:

$\frac{3(7m - 1)}{4} = \frac{21m - 3}{4}$

2. Simplify the second term $\frac{2m - 1 - m}{2}$:

$2m - 1 - m = m - 1$

Thus, the second term becomes:

$\frac{m - 1}{2}$

The LHS now becomes:

$\frac{21m - 3}{4} - \frac{m - 1}{2}$

Step 2: Find a common denominator

To simplify further, we need to rewrite both fractions with a common denominator. The least common denominator between 4 and 2 is 4.

$\frac{m - 1}{2} = \frac{2(m - 1)}{4} = \frac{2m - 2}{4}$

Thus, the equation becomes:

$\frac{21m - 3}{4} - \frac{2m - 2}{4} = m + \frac{3}{2}$

Step 3: Combine like terms on the LHS

Now that both terms on the LHS have the same denominator, we can combine them:

$\frac{(21m - 3) - (2m - 2)}{4} = \frac{21m - 3 - 2m + 2}{4} = \frac{19m - 1}{4}$

The equation now looks like:

$\frac{19m - 1}{4} = m + \frac{3}{2}$

Step 4: Eliminate the denominators by multiplying the entire equation by 4

To eliminate the fractions, multiply both sides of the equation by 4:

$19m - 1 = 4 \left(m + \frac{3}{2}\right)$

Step 5: Expand and solve for $m$

Now expand the right-hand side:

$4 \left(m + \frac{3}{2}\right) = 4m + 6$

Thus, the equation becomes:

$19m - 1 = 4m + 6$

Step 6: Solve for $m$

Now, subtract $4m$ from both sides:

$19m - 4m - 1 = 6$

Simplifying:

$15m - 1 = 6$

Now, add 1 to both sides:

$15m = 7$

Finally, divide by 15:

$m = \frac{7}{15}$

Final Answer:

$m = \frac{7}{15}$

Would you like more details or have any questions?

Related Questions:

1. How do you find a common denominator when dealing with fractions?
2. What are the steps to solve a linear equation with fractions?
3. How can you check if your solution is correct by substituting back into the original equation?
4. What happens if the equation contains variables on both sides?
5. How do you simplify expressions with multiple terms?

Tip:

Always try to clear fractions early in an equation by multiplying by the least common denominator—it simplifies the steps!