To solve the system of equations using the elimination method, we start by rewriting the equations:
- \( 0.1x + 0.3y = 0.9 \) (Equation 1)
- \( 0.1x - 0.3y = 0.2 \) (Equation 2)
Next, we can eliminate the decimals for simplicity. We can multiply both equations by 10 to eliminate the decimals:
- \( x + 3y = 9 \) (Equation 1 multiplied by 10)
- \( x - 3y = 2 \) (Equation 2 multiplied by 10)
Next, we can use the elimination method. We can subtract Equation 2 from Equation 1:
\[ (x + 3y) - (x - 3y) = 9 - 2 \]
Simplifying, we get:
\[ x + 3y - x + 3y = 7 \]
This simplifies to:
\[ 6y = 7 \]
Now, we can solve for \( y \):
\[ y = \frac{7}{6} \]
Now that we have \( y \), we can substitute it back into either of the original equations to solve for \( x \). Let's substitute \( y \) into Equation 1:
\[ x + 3\left(\frac{7}{6}\right) = 9 \]
This simplifies to:
\[ x + \frac{21}{6} = 9 \]
Now we can convert 9 to a fraction with a denominator of 6:
\[ x + \frac{21}{6} = \frac{54}{6} \]
Now, isolate \( x \):
\[ x = \frac{54}{6} - \frac{21}{6} = \frac{33}{6} = \frac{11}{2} \]
Thus, we have found \( x = \frac{11}{2} \) and \( y = \frac{7}{6} \).
The solution to the system of equations is:
\[ \left( \frac{11}{2}, \frac{7}{6} \right) \]
Therefore, the response from the provided options that matches our solution is:
left parenthesis Start Fraction 11 over 2 End Fraction comma Start Fraction 7 over 6 End Fraction right parenthesis.
1 answer