Problem 33 - Prealgebra

Question

Indicate which of the given ordered pairs are solutions for each equation. $$x+y=0 \quad(0,0),(5,-5),(-3,3)$$

Step-by-Step Solution

Answer

All ordered pairs \((0,0), (5,-5), (-3,3)\) are solutions to the equation.

Step 1: Understand the Equation

We are given the equation \(x + y = 0\). An ordered pair \((a, b)\) is a solution if substituting \(x = a\) and \(y = b\) satisfies the equation.

Step 2: Test the First Ordered Pair

Substitute \(x = 0\) and \(y = 0\) into the equation. So, we have: \(0 + 0 = 0\). This is true, so \((0,0)\) is a solution.

Step 3: Test the Second Ordered Pair

Substitute \(x = 5\) and \(y = -5\) into the equation. So, we have: \(5 + (-5) = 0\). This is true, so \((5,-5)\) is a solution.

Step 4: Test the Third Ordered Pair

Substitute \(x = -3\) and \(y = 3\) into the equation. So, we have: \(-3 + 3 = 0\). This is true, so \((-3,3)\) is a solution.

Key Concepts

Understanding Ordered Pairs

An ordered pair is a basic concept in mathematics that represents two elements, typically numbers, placed in a specific order, forming a couplet. The standard notation for an ordered pair is \((a, b)\). Here, \(a\) is known as the first element, or the "x-coordinate," and \(b\) is the second element, or the "y-coordinate." These pairs are fundamental in graphing on the coordinate plane, where the order dictates a specific point's location.

In the context of linear equations, ordered pairs are crucial because they help us identify coordinates that can be substituted into an equation to verify the solutions to that equation. For instance, given an equation like \(x + y = 0\), we can use ordered pairs to see which values of \(x\) and \(y\) satisfy this condition.

Practically, each pair represents a positional relationship between the two variables. Ordered pairs offer a precise way to check whether a particular combination of values is valid for a given equation.

Identifying Solutions of Equations

A solution to an equation in mathematics refers to a value or set of values that, when substituted, make an equation true. For linear equations like \(x + y = 0\), solutions are the paired values of \(x\) and \(y\) which satisfy the equation.

To determine whether certain ordered pairs are solutions, you replace the variable values with those from the pair and perform the arithmetic to check if the original equation holds. For example:

• For the equation \(x + y = 0\), when checking the ordered pair \((0, 0)\), substitute to get \(0 + 0 = 0\), which ensures it is a solution.
• In the same way, the pair \((5, -5)\) gives \(5 + (-5) = 0\), making it a solution.
• Lastly, the pair \((-3, 3)\) results in \(-3 + 3 = 0\), confirming it is also a solution.

An ordered pair is thus a solution if substituting its elements into the equation results in a true statement. This process is essential for verifying potential solutions among multiple candidates.

The Art of Testing Solutions

To test whether an ordered pair is a solution to a given equation, we perform substitution and evaluate the equation to see if it holds true. This checking process stands at the heart of ensuring an equation's validity with specific values.

Here's how you can effectively test solutions:

• **Substitute**: Begin by substituting the first element of your ordered pair into the variable \(x\), and the second element into \(y\). This direct replacement aligns the pair's values with the equation.
• **Calculate**: Perform the necessary arithmetic operations after substitution. Be sure to simplify the expression fully to determine the outcome.
• **Check**: Compare the result of your calculations with the right side of the equation. If both sides of the equation are equal after the computation, the ordered pair is confirmed as a solution.

Testing solutions by substitution and calculation is a reliable technique to verify whether a pair qualifies as a solution, hence grounding your understanding in concrete examples.