How can we solve the system of equations represented by 2x + y = 3 and x + 2y = 1?

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Solving the System of Equations

To solve the system of equations
2x + y = 3
and
x + 2y = 1, we can use the method of substitution or elimination. Here, we’ll use the elimination method for clarity.

Step 1: Align the equations

First, we write the equations one under the other:

    2x +  y = 3  (Equation 1)
    x + 2y = 1  (Equation 2)

Step 2: Eliminate one variable

Let’s eliminate y by making the coefficients of y in both equations the same. To do this, we can multiply Equation 1 by 2:

    4x + 2y = 6  (Transformed Equation 1)
    x + 2y = 1   (Equation 2)

Now we have:

    4x + 2y = 6  (Transformed Equation 1)
    x + 2y = 1   (Equation 2)

Step 3: Subtract the equations

Next, we subtract Equation 2 from the Transformed Equation 1:

    (4x + 2y) - (x + 2y) = 6 - 1

This simplifies to:

    3x = 5

Now, divide by 3:

    x = 5/3

Step 4: Substitute back to find y

Now that we have x, we can substitute back into either original equation to find y. Let’s use Equation 1:

    2(5/3) + y = 3

This simplifies to:

    10/3 + y = 3

To isolate y, subtract 10/3 from both sides:

    y = 3 - 10/3

Converting 3 to a fraction gives us:

    y = 9/3 - 10/3 = -1/3

Final Solution

The solution to the system of equations is:

    x = 5/3
    y = -1/3

Thus, the pair that satisfies both equations is (5/3, -1/3).

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