To solve the system of equations given by
- \( 0.5x - 1.4y = 5.8 \)
- \( y = -0.3x - \frac{1}{5} \)
we can substitute the expression for \( y \) from the second equation into the first equation.
Step 1: Substitute for \( y \)
Substituting \( y \) in the first equation:
\[ 0.5x - 1.4(-0.3x - \frac{1}{5}) = 5.8 \]
Step 2: Simplify the equation
Distributing \( -1.4 \):
\[ 0.5x + 1.4(0.3)x + 1.4(\frac{1}{5}) = 5.8 \]
Calculating \( 1.4 \times 0.3 \):
\[ 0.5x + 0.42x + 1.4(\frac{1}{5}) = 5.8 \]
Calculating \( 1.4 \times \frac{1}{5} = \frac{1.4}{5} = 0.28 \):
\[ 0.5x + 0.42x + 0.28 = 5.8 \]
Step 3: Combine like terms
Combine \( 0.5x \) and \( 0.42x \):
\[ (0.5 + 0.42)x + 0.28 = 5.8 \]
This simplifies to:
\[ 0.92x + 0.28 = 5.8 \]
Step 4: Solve for \( x \)
Subtract \( 0.28 \) from both sides:
\[ 0.92x = 5.8 - 0.28 \] \[ 0.92x = 5.52 \]
Now, divide by \( 0.92 \):
\[ x = \frac{5.52}{0.92} \]
Calculating the above division:
\[ x = 6 \]
Step 5: Solve for \( y \)
Now, substitute \( x = 6 \) back into the second equation to find \( y \):
\[ y = -0.3(6) - \frac{1}{5} \] \[ y = -1.8 - 0.2 \] \[ y = -2.0 \]
Final Solution
The solution to the system of equations is:
\[ (x, y) = (6, -2.0) \]
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