Problem 34 - 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

The solutions are (0,0) and (3,3).

Step 1: Understanding the Equation

First, we understand the equation given: \(x - y = 0\). This implies that for any ordered pair \((x, y)\), the value of \(x\) must equal the value of \(y\) for the pair to be a solution to the equation.

Step 2: Testing the Pair (0,0)

Evaluate the ordered pair \((0,0)\). Substitute \(x = 0\) and \(y = 0\) into the equation: \(0 - 0 = 0\). Since this statement is true, (0,0) is a solution.

Step 3: Testing the Pair (5,-5)

Evaluate the ordered pair \((5,-5)\). Substitute \(x = 5\) and \(y = -5\) into the equation: \(5 - (-5) = 5 + 5 = 10\). Since this statement is not zero, (5,-5) is not a solution.

Step 4: Testing the Pair (3,3)

Evaluate the ordered pair \((3,3)\). Substitute \(x = 3\) and \(y = 3\) into the equation: \(3 - 3 = 0\). Since this statement is true, (3,3) is a solution.

Key Concepts

Understanding Ordered Pairs

In mathematics, an ordered pair is a fundamental concept used to represent a pair of coordinated values. It is denoted as \((x, y)\), where the first element is the 'x-value' and the second element is the 'y-value.'

• Ordered pairs are used to show the relationship between two values, usually in a two-dimensional space or for solving equations.
• In the context of the equation \(x - y = 0\), a solution is an ordered pair where the 'x-value' equals the 'y-value.'

It's important to note the order—\((x, y)\) is not the same as \((y, x)\). This means for equations, precision in the order makes all the difference when determining a solution.Understanding ordered pairs helps in visualizing equations as points on a graph and verifying which points satisfy a given linear equation.

Explaining Solution Verification

Solution verification is the method used to check if an ordered pair satisfies a given equation. For a pair \((x, y)\) to be a solution of an equation, substituting these values into the equation must result in a true statement.To verify a solution:

• Identify the equation you need to solve. In this case, it is \(x - y = 0\).
• Take the ordered pair \((x, y)\) and substitute 'x' and 'y' into the equation.
• Simplify the equation after substitution and check if both sides of the equation are equal.

For example, to check if \((3, 3)\) is a solution:

• Substitute \(x = 3\) and \(y = 3\) into the equation: \(3 - 3 = 0\).
• The statement is true, so \((3, 3)\) is a solution.

This process steps through each ordered pair, ensuring careful evaluation of potential solutions.

Introducing the Substitution Method

The substitution method is a technique used primarily in solving systems of equations, but it's applicable to verify or find solutions for individual equations as well.Here's how to use the substitution method:

• Start with one equation, like \(x - y = 0\).
• Isolate one variable on one side of the equation. For this equation, it's already isolated as \(x = y\).
• Select known values from an ordered pair and substitute them back into the equation to simplify and verify.

For the pair \((0, 0)\):1. Substitute \(x = 0\) and \(y = 0\) into the equation: \(0 - 0 = 0\).2. Both sides of the equation equal zero, confirming \((0, 0)\) is a correct solution.The substitution method is especially useful in complex equations where you solve for one variable in terms of the other before identifying certainly valid solutions.