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

There is no solution
There is no solution

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

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

(−76,112)

1 answer

To solve the system of equations using the elimination method, we start with the given equations:

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

First, let's eliminate one of the variables, say \( y \). We can do this by adding the two equations together.

Step 1: Add the equations together:

\[ (0.1x + 0.3y) + (0.1x - 0.3y) = 0.9 + 0.2 \]

This simplifies to:

\[ 0.2x + 0y = 1.1 \]

Thus, we have:

\[ 0.2x = 1.1 \]

Step 2: Solve for \( x \):

To solve for \( x \), divide both sides by \( 0.2 \):

\[ x = \frac{1.1}{0.2} = 5.5 \]

Step 3: Substitute \( x \) back into one of the original equations to find \( y \):

We can substitute \( x = 5.5 \) into Equation 1:

\[ 0.1(5.5) + 0.3y = 0.9 \]

Calculating \( 0.1(5.5) \):

\[ 0.55 + 0.3y = 0.9 \]

Now, solve for \( 0.3y \):

\[ 0.3y = 0.9 - 0.55 \]

\[ 0.3y = 0.35 \]

Step 4: Solve for \( y \):

\[ y = \frac{0.35}{0.3} = \frac{35}{30} = \frac{7}{6} \]

Now we have the solution:

\[ (x, y) = (5.5, \frac{7}{6}) \]

In fractional form, \( 5.5 = \frac{11}{2} \).

Therefore, the solution to the system of equations is:

\[ \left( \frac{11}{2}, \frac{7}{6} \right) \]

The correct response option is:

\(\left( \frac{11}{2}, \frac{7}{6} \right)\)