`{:(0.4x + 0.3y = 1.7),(0.7x - 0.2y = 0.8):}`

Answer

To solve the system of equations given by:

1. \(0.4x + 0.3y = 1.7\) (Equation 1)
2. \(0.7x - 0.2y = 0.8\) (Equation 2)

we will follow these steps:

### Step 1: Eliminate the decimals
To make calculations easier, we can multiply both equations by 10 to eliminate the decimals.

- For Equation 1:
\[ 10(0.4x + 0.3y) = 10(1.7) \implies 4x + 3y = 17 \]

- For Equation 2:
\[ 10(0.7x - 0.2y) = 10(0.8) \implies 7x - 2y = 8 \]

Now we have the new system of equations:
1. \(4x + 3y = 17\) (Equation 3)
2. \(7x - 2y = 8\) (Equation 4)

### Step 2: Solve the equations
Now we will solve these equations using the elimination method.

#### Adding the equations:
We can add Equation 3 and Equation 4:
\[ (4x + 3y) + (7x - 2y) = 17 + 8 \]
This simplifies to:
\[ 11x + y = 25 \quad \text{(Equation 5)} \]

#### Subtracting the equations:
Next, we will subtract Equation 4 from Equation 3:
\[ (4x + 3y) - (7x - 2y) = 17 - 8 \]
This simplifies to:
\[ -3x + 5y = 9 \quad \text{(Equation 6)} \]

### Step 3: Solve for one variable
Now we will solve for \(y\) in terms of \(x\) using Equation 5:
\[ y = 25 - 11x \quad \text{(Equation 7)} \]

### Step 4: Substitute into another equation
Now we will substitute Equation 7 into Equation 6:
\[ -3x + 5(25 - 11x) = 9 \]
Expanding this gives:
\[ -3x + 125 - 55x = 9 \]
Combining like terms:
\[ -58x + 125 = 9 \]
Subtracting 125 from both sides:
\[ -58x = 9 - 125 \]
\[ -58x = -116 \]
Dividing by -58:
\[ x = \frac{-116}{-58} = 2 \]

### Step 5: Find the value of \(y\)
Now we substitute \(x = 2\) back into Equation 7 to find \(y\):
\[ y = 25 - 11(2) \]
\[ y = 25 - 22 = 3 \]

### Final Solution
Thus, the solution to the system of equations is:
\[ x = 2, \quad y = 3 \]