solve the following system of equations using the elimination method. 2x + y = 3.5

-x + 2y = 2.5

1 answer

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)

First, we'll manipulate these equations so we can eliminate one of the variables.

Let's manipulate Equation 1 to make the coefficients of \(y\) the same in both equations. We can multiply Equation 1 by 2:

\[ 2(2x + y) = 2(3.5) \] This gives us:

\[ 4x + 2y = 7 \quad \text{(Equation 3)} \]

Now we have the modified system:

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

Next, we'll eliminate \(y\) by subtracting Equation 2 from Equation 3:

\[ (4x + 2y) - (-x + 2y) = 7 - 2.5 \]

Simplifying this, we get:

\[ 4x + 2y + x - 2y = 7 - 2.5 \] \[ 5x = 4.5 \]

Now, solving for \(x\):

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

Now that we have \(x\), we can substitute this value back into one of the original equations to find \(y\). Let's use Equation 1:

\[ 2(0.9) + y = 3.5 \]

This simplifies to:

\[ 1.8 + y = 3.5 \]

Now, solving for \(y\):

\[ y = 3.5 - 1.8 = 1.7 \]

So, the solution to the system of equations is:

\[ x = 0.9, \quad y = 1.7 \]

Thus, the solution in ordered pair form is:

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

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