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An ordered pair is a set of two numbers written in a specific order, usually as \(x, y\). In mathematics, ordered pairs are crucial as they represent points on the coordinate plane. Each number corresponds to one axis: the first number, \(x\), indicates the horizontal position, while the second number, \(y\), reflects the vertical position.

Ordered pairs help in plotting points on graphs, where the intersection of \(x\) and \(y\) values gives us a precise location. For example, in the ordered pair \((1, 1)\), \(x = 1\) and \(y = 1\). This tells us to start at the origin (0,0), move 1 unit along the \(x\)-axis, and then move 1 unit up along the \(y\)-axis. This also serves as a potential solution to an equation when substituting the values into the equation confirms that the statement is true.

Verifying a solution in the context of equations means checking whether a given ordered pair satisfies the equation. To do this, you substitute the \(x\) and \(y\) values from the ordered pair into the equation and see if you end up with a valid identity. Here’s how it works:

• Substitute the \(x\) value from the ordered pair into the equation wherever you see \(x\).
• Do the same with the \(y\) value, substituting it wherever \(y\) appears in the equation.
• Simplify both sides of the equation to check if they are equal.

For instance, in the equation \(y = 5x - 4\) with the ordered pair \((1, 1)\), substitute \(x = 1\) and \(y = 1\), resulting in \(1 = 5(1) - 4\). Simplifying gives \(1 = 1\), verifying that the ordered pair is indeed a solution.

The substitution method is a straightforward approach used to verify if an ordered pair satisfies an equation. It involves plugging in the values from the ordered pair directly into the equation.

Using substitution involves the following steps:

• Identify the variables and their values from the ordered pair. In \((x, y) = (1, 1)\), \(x = 1\) and \(y = 1\).
• Insert these values into the equation where \(x\) and \(y\) occur.
• Calculate or simplify the equation to see if it holds true.

This method helps confirm the solution quickly by verifying that both sides of the equation are equal, as demonstrated in the equation \(y = 5x - 4\) becoming \(1 = 5(1) - 4\), which simplifies to true as \(1 = 1\).

Mastering substitution can greatly enhance your algebraic problem-solving skills.