Solving equations with two variables builds directly on the concept of solving one-variable equations. In a one-variable equation, such as x + 2 = 5, a value for x is a solution if substituting it into the equation results in a true statement. For example, plugging in x = 3 gives 3 + 2 = 5, which is true, so x = 3 is a solution.

When working with two-variable equations, the principle is similar but involves ordered pairs in the form (x, y). An ordered pair is a solution if substituting x and y into the equation makes it true. For instance, consider the linear equation x + 2y = 5. To check if the ordered pair (3, 1) is a solution, substitute x = 3 and y = 1:

$$3 + 2 \times 1 = 3 + 2 = 5$$

Since the left side equals the right side, the ordered pair (3, 1) satisfies the equation.

To determine whether other ordered pairs are solutions, substitute their values into the equation and verify if the equality holds. For example, with the pair (5, 0):

$$5 + 2 \times 0 = 5 + 0 = 5$$

This is true, so (5, 0) is a solution. However, for the pair (8, 2):

$$8 + 2 \times 2 = 8 + 4 = 12 \neq 5$$

This is false, so (8, 2) is not a solution. Lastly, for the pair (-3, 4):

$$-3 + 2 \times 4 = -3 + 8 = 5$$

This is true, confirming that (-3, 4) is a solution.

Understanding how to verify solutions for two-variable equations is essential for solving systems of equations and analyzing linear relationships. The key is to substitute the values of x and y into the equation and check if the resulting statement is true. This method extends the foundational skills from one-variable equations to more complex scenarios involving multiple variables.

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