Question

Solve the following system of equations using the elimination method.

0.1x+0.3y=0.9
, 0.1x−0.3y=0.2

(1 point)
Responses

(112,−76)
left parenthesis Start Fraction 11 over 2 End Fraction comma negative Start Fraction 7 over 6 End Fraction right parenthesis

(−76,112)
left parenthesis negative Start Fraction 7 over 6 End Fraction comma Start Fraction 11 over 2 End Fraction right parenthesis

There is no solution
There is no solution

(112,76)

Answer 1

To solve the system of equations using the elimination method, we start by rewriting the equations:

1. \( 0.1x + 0.3y = 0.9 \) (Equation 1)
2. \( 0.1x - 0.3y = 0.2 \) (Equation 2)

Next, we can eliminate the decimals for simplicity. We can multiply both equations by 10 to eliminate the decimals:

1. \( x + 3y = 9 \) (Equation 1 multiplied by 10)
2. \( x - 3y = 2 \) (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**.