To solve the system of equations using the substitution method, we start with the two equations:
- \( 0.2x - 1.9y = -1.4 \) (Equation 1)
- \( -0.1x + 2.5y = 3.8 \) (Equation 2)
Step 1: Solve one of the equations for one variable
Let's solve Equation 1 for \( x \):
\[ 0.2x = 1.9y - 1.4 \] \[ x = \frac{1.9y - 1.4}{0.2} \] \[ x = 9.5y - 7 \]
Step 2: Substitute into the other equation
Now substitute \( x \) in Equation 2:
\[ -0.1(9.5y - 7) + 2.5y = 3.8 \]
Distributing \(-0.1\):
\[ -0.95y + 0.7 + 2.5y = 3.8 \]
Combine like terms:
\[ (-0.95y + 2.5y) + 0.7 = 3.8 \] \[ 1.55y + 0.7 = 3.8 \]
Step 3: Solve for \( y \)
Subtract \( 0.7 \) from both sides:
\[ 1.55y = 3.8 - 0.7 \] \[ 1.55y = 3.1 \]
Now divide both sides by \( 1.55 \):
\[ y = \frac{3.1}{1.55} = 2 \]
Step 4: Substitute back to find \( x \)
Now substitute \( y = 2 \) back into the equation we derived for \( x \):
\[ x = 9.5(2) - 7 \] \[ x = 19 - 7 \] \[ x = 12 \]
Step 5: Check the solution
We found \( x = 12 \) and \( y = 2 \). Now we will check these values in both original equations.
Check Equation 1:
\[ 0.2(12) - 1.9(2) = -1.4 \] \[ 2.4 - 3.8 = -1.4 \quad \text{(True)} \]
Check Equation 2:
\[ -0.1(12) + 2.5(2) = 3.8 \] \[ -1.2 + 5 = 3.8 \quad \text{(True)} \]
Both equations are satisfied.
Conclusion
The solution to the system of equations is:
\[ \boxed{(12, 2)} \]