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

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)

Answer 1

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:**

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