The equation you need to solve is:

$r - 2 = \frac{-2}{5} + \frac{7}{10} + \frac{3}{2}$

Let's solve this step-by-step.

Step 1: Simplify the right-hand side

We need to combine the fractions on the right-hand side. To do this, we find the least common denominator (LCD) of the fractions.

• The denominators are 5, 10, and 2.
• The LCD of 5, 10, and 2 is 10.

Now, rewrite each fraction with the denominator of 10: $\frac{-2}{5} = \frac{-4}{10}, \quad \frac{7}{10} = \frac{7}{10}, \quad \frac{3}{2} = \frac{15}{10}$

Step 2: Add the fractions

Now, add the fractions: $\frac{-4}{10} + \frac{7}{10} + \frac{15}{10} = \frac{-4 + 7 + 15}{10} = \frac{18}{10}$ Simplify $\frac{18}{10}$: $\frac{18}{10} = 1.8$

Step 3: Solve for $r$

Now, substitute the simplified value back into the equation: $r - 2 = 1.8$

Add 2 to both sides: $r = 1.8 + 2 = 3.8$

Thus, the solution is: $r = 3.8$

Would you like more details or have any questions about this process?

Related Questions:

1. How do you find the least common denominator of fractions?
2. How can you simplify fractions like $\frac{18}{10}$?
3. What are some methods to solve linear equations with fractions?
4. How can you check your solution to a linear equation?
5. What happens if you have variables on both sides of the equation?

Tip: Always simplify fractions before adding or subtracting them to avoid errors.