Let a equal `2b + 3c - 5`.What happens to the value of a if the value of b decreases by 1 and the value of c increases by 2?

Options

A. It increases by 4
B. It is increases by 2
C. It increases by 1
D. It is unchanged

Answer

To solve the problem step by step, we will analyze how the value of \( a \) changes when \( b \) decreases by 1 and \( c \) increases by 2.

### Step 1: Write the original expression for \( a \)
The original expression given is:
\[ a = 2b + 3c - 5 \]

**Hint:** Identify the initial formula that relates \( a \), \( b \), and \( c \).

### Step 2: Substitute the changes in \( b \) and \( c \)
We are told that \( b \) decreases by 1 and \( c \) increases by 2. Therefore, we substitute \( b - 1 \) for \( b \) and \( c + 2 \) for \( c \) in the expression for \( a \):
\[ a' = 2(b - 1) + 3(c + 2) - 5 \]

**Hint:** Make sure to replace \( b \) and \( c \) with their new values correctly.

### Step 3: Expand the new expression for \( a' \)
Now, we will expand the expression:
\[ a' = 2(b - 1) + 3(c + 2) - 5 \]
\[ = 2b - 2 + 3c + 6 - 5 \]

**Hint:** Distribute the coefficients and combine like terms carefully.

### Step 4: Combine like terms
Now, let's combine the constant terms:
\[ a' = 2b + 3c - 2 + 6 - 5 \]
\[ = 2b + 3c - 2 + 1 \]
\[ = 2b + 3c + 4 \]

**Hint:** Keep track of the constants to ensure they are combined correctly.

### Step 5: Compare the new expression \( a' \) with the original expression \( a \)
The original expression for \( a \) was:
\[ a = 2b + 3c - 5 \]
The new expression we derived is:
\[ a' = 2b + 3c + 4 \]

### Step 6: Determine the change in \( a \)
To find out how much \( a \) has changed, we compare \( a' \) with \( a \):
\[ a' - a = (2b + 3c + 4) - (2b + 3c - 5) \]
\[ = 4 + 5 \]
\[ = 9 \]

**Hint:** The change in \( a \) is the difference between the new and original expressions.

### Conclusion
Thus, the value of \( a \) increases by \( 9 \).

### Final Answer
The value of \( a \) increases by \( 9 \).