How can we solve the system of equations 2x + 2y = 8 and x + 2y = 1 using the elimination method?

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

  • Equation 1: 2x + 2y = 8
  • Equation 2: x + 2y = 1

The goal of the elimination method is to manipulate the equations such that one variable can be eliminated when we add or subtract the equations. Here’s how we can proceed:

Step 1: Simplify the Equations

First, we can simplify Equation 1 by dividing every term by 2:

  • Equation 1 becomes: x + y = 4

Now we have:

  • Equation 1: x + y = 4
  • Equation 2: x + 2y = 1

Step 2: Align the Equations for Elimination

Next, we need to eliminate one of the variables. We’ll eliminate x by subtracting Equation 1 from Equation 2:

  • (x + 2y) – (x + y) = 1 – 4

This simplifies to:

  • y = -3

Step 3: Substitute Back to Find x

Now that we have the value of y, we can substitute it back into one of the original equations to find x. We can use Equation 1:

  • x + (-3) = 4

Solving for x, we get:

  • x = 4 + 3 = 7

Step 4: State the Solution

The solution to the system of equations is:

  • x = 7
  • y = -3

Thus, the coordinates of the solution are (7, -3).

Summary:

Using the elimination method, we simplified the original system and eliminated one variable to find:

Solution: (7, -3)

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