What is the simplest method to solve the system of equations 7x + 5y = 19 and 7x + 2y = 16?

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To solve the system of equations 7x + 5y = 19 and 7x + 2y = 16, one of the simplest methods you can use is the elimination method. This method allows us to eliminate one variable, making it easier to solve for the other.

Here are the steps to apply the elimination method:

  1. Write down the equations:
    • Equation 1: 7x + 5y = 19
    • Equation 2: 7x + 2y = 16
  2. Eliminate one variable: Notice that both equations have the term 7x. To eliminate 7x, we can subtract Equation 2 from Equation 1:

    3. (7x + 5y) – (7x + 2y) = 19 – 16

  3. Perform the subtraction:

    4. This simplifies to:

    (7x – 7x) + (5y – 2y) = 3

    Which further simplifies to:

    3y = 3

  4. Solve for y:

    5. Dividing both sides by 3 gives us:

    y = 1

  5. Substitute y back into one of the original equations: We can use Equation 1 for this purpose:

    6. 7x + 5(1) = 19

    Which simplifies to:

    7x + 5 = 19

  6. Solve for x:

    7. Subtracting 5 from both sides yields:

    7x = 14

    Dividing both sides by 7 gives us:

    x = 2

Final Solution: The solution to the system of equations is x = 2 and y = 1. This means that the point of intersection for these two lines is (2, 1).

In conclusion, the elimination method is particularly effective in this case because it allows for a straightforward way to solve for one variable quickly. By taking advantage of the structure of these equations, you can efficiently find the solution.

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