To solve the system of equations given by:
- \(0.2x + 0.7y = 2.2\)
- \(5x - 3y = 14\)
we will use the elimination method.
Step 1: Eliminate a variable
First, let's eliminate \(y\). We can do this by aligning the coefficients for \(y\).
We want to make the coefficients of \(y\) in both equations the same. The first equation has a coefficient of \(0.7\) for \(y\).
To compare, we can multiply the first equation by \(3\) to get the coefficients of \(y\) in both equations compatible:
\[ 3(0.2x + 0.7y) = 3(2.2) \]
This simplifies to:
\[ 0.6x + 2.1y = 6.6 \quad \text{(Equation 3)} \]
Step 2: Align the equations
Now rewrite the second equation \(5x - 3y = 14\) (Equation 2) to prepare for elimination:
\[ 5x - 3y = 14 \quad \text{(Equation 2)} \]
Step 3: Make the coefficients of \(y\) match
Next, we will multiply the second equation by \(0.7\) to get the coefficients of \(y\) to match:
\[ 0.7(5x - 3y) = 0.7(14) \]
Which simplifies to:
\[ 3.5x - 2.1y = 9.8 \quad \text{(Equation 4)} \]
Step 4: Set up for elimination
Now we have the following system:
- \(0.6x + 2.1y = 6.6\) (Equation 3)
- \(3.5x - 2.1y = 9.8\) (Equation 4)
Step 5: Add the equations to eliminate \(y\)
Add Equation 3 and Equation 4:
\[ (0.6x + 2.1y) + (3.5x - 2.1y) = 6.6 + 9.8 \]
This simplifies to:
\[ 4.1x = 16.4 \]
Step 6: Solve for \(x\)
Now, divide both sides by \(4.1\):
\[ x = \frac{16.4}{4.1} = 4 \]
Step 7: Substitute back to find \(y\)
Now substitute \(x = 4\) back into one of the original equations. We'll use the first equation:
\[ 0.2(4) + 0.7y = 2.2 \]
This simplifies to:
\[ 0.8 + 0.7y = 2.2 \]
Subtract \(0.8\) from both sides:
\[ 0.7y = 1.4 \]
Now divide by \(0.7\):
\[ y = \frac{1.4}{0.7} = 2 \]
Final Solution:
Thus, the solution to the system of equations is:
\[ x = 4, \quad y = 2 \]
So, the answer is: A. The solution of the system is \(x = 4\) and \(y = 2\).
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