Question

Solve the following system of equations using the elimination method. 2x + y = 3.5 , −x + 2y = 2.5

Answer 1

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

1. \( 2x + y = 3.5 \) (Equation 1)
2. \( -x + 2y = 2.5 \) (Equation 2)

Our goal is to eliminate one of the variables by manipulating these equations. Let's eliminate \( y \).

First, we can make the coefficients of \( y \) the same. To do this, we can multiply Equation 1 by 2:

\[
2(2x + y) = 2(3.5)
\]
This simplifies to:

\[
4x + 2y = 7 \tag{Equation 3}
\]

Now we have:

1. \( 4x + 2y = 7 \) (Equation 3)
2. \( -x + 2y = 2.5 \) (Equation 2)

Next, we will subtract Equation 2 from Equation 3 to eliminate \( y \):

\[
(4x + 2y) - (-x + 2y) = 7 - 2.5
\]
This leads to:

\[
4x + 2y + x - 2y = 7 - 2.5
\]
\[
5x = 4.5
\]
Now, we solve for \( x \):

\[
x = \frac{4.5}{5} = 0.9
\]

Now that we have \( x \), we can substitute \( x = 0.9 \) back into one of the original equations to solve for \( y \). We can use Equation 1:

\[
2(0.9) + y = 3.5
\]
This simplifies to:

\[
1.8 + y = 3.5
\]
Now, solve for \( y \):

\[
y = 3.5 - 1.8 = 1.7
\]

Thus, the solution to the system of equations is:

\[
\boxed{(0.9, 1.7)}
\]