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An ordered pair is a pair of numbers used to locate a point on a coordinate plane. It's written in the form \(x, y\), where \(x\) is the horizontal position, and \(y\) is the vertical position. Each element of an ordered pair represents a coordinate on a two-dimensional plane. Think of it as an address that tells you where to find a point.

Ordered pairs are essential for plotting graphs and solving equations because they show the exact location of a point in space. For example, if you have an ordered pair like \((3, -2)\), it means you move 3 units right (since \(x\) is positive) and 2 units down (since \(y\) is negative) to reach the point.

When completing an ordered pair for a given equation, you usually solve for either \(x\) or \(y\) so that both fit the conditions imposed by the equation. This ensures that the pair is on the graph of the equation.

The substitution method is a powerful way to solve systems of equations, especially linear ones. It involves replacing one variable with an equivalent expression from another equation. This is useful when you have multiple equations and want to find a solution that satisfies all of them.

In our context, the substitution method works well for completing ordered pairs. Here's how it was applied to our example exercise:

• Start by identifying the value given in the ordered pair—in this case, the \(y\)-value (-2).
• Substitute this \(y\)-value into the equation to find the corresponding \(x\)-value.
• Solve the modified equation to isolate the remaining variable.

By using the substitution method, you simplify a problem that could seem tricky at first. It's a systematic approach that ensures you find consistent solutions.

Solving for variables is a crucial step in algebra, helping you find the unknowns in an equation. In linear equations, you usually have one or more variables that you need to isolate and solve for their values. Here's a simple approach to solving for variables in linear equations:

• First, simplify the equation wherever possible by combining like terms or distributing constants.
• Next, focus on isolating the variable of interest. This often involves moving terms that don't contain the variable to the other side of the equation through addition or subtraction.
• Once the variable terms are on one side, divide or multiply to get the variable by itself, solving the equation.

In our exercise, we isolated \(x\) to complete the ordered pair:

• Started with \((5x + 18 = 3)\),
• Subtracted 18 to get the \(5x = -15\),
• Finally, divided by 5, resulting in \(x = -3\).

By mastering solving for variables, you'll handle many algebraic problems efficiently!