What must be added to `5a , -3b + 2c " to get " 3a - 4b + 7c `?

Answer

To solve the problem of what must be added to \(5a - 3b + 2c\) to get \(3a - 4b + 7c\), we can follow these steps:

### Step 1: Set up the equation
We need to find an expression \(x\) such that:
\[ 5a - 3b + 2c + x = 3a - 4b + 7c \]

### Step 2: Rearrange the equation
To isolate \(x\), we can rearrange the equation:
\[ x = (3a - 4b + 7c) - (5a - 3b + 2c) \]

### Step 3: Distribute the negative sign
Now, we distribute the negative sign across the second expression:
\[ x = 3a - 4b + 7c - 5a + 3b - 2c \]

### Step 4: Combine like terms
Now, we will combine the like terms:
- For the \(a\) terms: \(3a - 5a = -2a\)
- For the \(b\) terms: \(-4b + 3b = -1b\) (or simply \(-b\))
- For the \(c\) terms: \(7c - 2c = 5c\)

Putting it all together, we have:
\[ x = -2a - b + 5c \]

### Step 5: Conclusion
Thus, the expression that must be added to \(5a - 3b + 2c\) to obtain \(3a - 4b + 7c\) is:
\[ -2a - b + 5c \]

---