Equations express relationships between variables and constants. The solutions to two-variable equations consist of two values, known as ordered pairs, and written as (a, b) where "a" and "b" are real-number constants. An equation can have an infinite number of ordered pairs that make the original equation true. Ordered pairs are useful for plotting the graph of an equation.

Rewrite the equation in terms of one of the variables. Note that terms change signs when they move from one side of an equation to another. For example, rewrite y - x^2 + 2x = 5 as y = x^2 - 2x + 5.

Construct a two-column table, also known as a T-table, for the ordered pairs. Label the columns "x" and "y" for the two variables. Write positive and negative values for "x" and solve for the corresponding values of "y." In the example, use values of -1, 0 and 1 for “x” to start the table. The corresponding y-values are y = (-1)^2 - 2(-1) + 5 = 8, y = 0 - 0 + 5 = 5 and y = (1)^2 - 2(1) + 5 = 4. So the first three ordered pair solutions are (-1, 8), (0, 5) and (1, 4). You can plot these first few points to get a preliminary idea of the shape of the curve.

Find the ordered pair for a system of equations. A simple way to solve a two-equation system is to try to eliminate one of the variable terms, add the two equations and then solve for both variables. For example, if you have two equations, 2x + 3y = 5 and x - y = 5, multiply the second equation by -2 to get -2x + 2y = -10. Now, add the two equations to get 2x + 3y - 2x + 2y = 5 – 10, which simplifies to 5y = -5, or y = -1. Substitute the “y” value into either one of the original equations to solve for “x.” So x - (-1) = 5, which simplifies to x + 1 = 5, or x = 4. So the ordered pair that makes both equations true is (4, -1). Note that not all equation systems may have solutions.

Verify if an ordered pair satisfies an equation. Substitute either the x- or the y-value from the ordered pair and see if the equation is satisfied. In the example, examine whether the ordered pair (2, 1) make the equation y = x^2 - 2x + 5 true. Substituting x = 2 into the equation, you get y = (2)^2 - 2(2) + 5 = 4 - 4 + 5. So the ordered pair (2, 1) is not a solution of the equation. For a system of equations, substitute the ordered pair in each equation to see if they are made true.

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About the Author

Based in Ottawa, Canada, Chirantan Basu has been writing since 1995. His work has appeared in various publications and he has performed financial editing at a Wall Street firm. Basu holds a Bachelor of Engineering from Memorial University of Newfoundland, a Master of Business Administration from the University of Ottawa and holds the Canadian Investment Manager designation from the Canadian Securities Institute.

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