Problem 41 - Introductory and Intermediate Algebra for College Students 4th

Question

Determine whether each ordered pair is a solution of the given equation. $$y=2 x+6 \quad(0,6),(-3,0),(2,-2)$$

Step-by-Step Solution

Answer

The ordered pairs (0,6) and (-3,0) are solutions to the equation \(y = 2x + 6\), but the pair (2,-2) is not.

Step 1: Substitute the first ordered pair

Begin by replacing x and y in the equation with the values from the first ordered pair (0,6). Here, x = 0 and y = 6, so the equation becomes: \(6 = 2(0) + 6\). By simplifying this, it becomes \(6 = 6\), which is true. This implies that the first ordered pair (0,6) is a solution to the equation.

Step 2: Substitute the second ordered pair

Next, substitute x and y in the equation with the values from the second ordered pair, which are (-3,0). So, the equation becomes: \(0 = 2(-3)+6\). By simplifying this, it becomes \(0 = 0\), which is true. Thus, the second ordered pair (-3,0) is a solution to the equation.

Step 3: Substitute the third ordered pair

Lastly, substitute x and y in the equation with the values from the third ordered pair (2,-2). Now, the equation becomes: \(-2 = 2(2) + 6\). After simplifying this, we get \(-2 = 10\), which is false. So, the third pair (2,-2) is not a solution to the equation.

Key Concepts

Understanding Ordered Pairs and Linear Equations

Ordered pairs are a fundamental concept when working with linear equations. They are expressed in the form \((x, y)\), where \(x\) is the value on the horizontal axis and \(y\) on the vertical axis.
When given a linear equation such as \(y = 2x + 6\), we can determine whether an ordered pair fits the equation by substituting \(x\) and \(y\) with the coordinates of the ordered pair.
If both sides of the equation remain equal after substitution, it confirms the pair is a solution.
Understanding this, we can test several pairs to see if they lie on the line represented by the equation.

• The point \((0, 6)\) lies on the line since substituting gives \(6 = 6\).
• So does \((-3, 0)\) because substituting results in \(0 = 0\).
• However, \((2, -2)\) is not a solution as it results in \(-2 = 10\).

Mastering ordered pairs in linear equations is essential for graphing lines and solving more complex equations.

Solution Verification Techniques

Solution verification is a crucial step in solving mathematical problems. Especially in testing if an ordered pair satisfies a linear equation. The process involves several steps:

• First, take the given ordered pair and substitute \(x\) and \(y\) into the equation.
• Compute each side of the equation separately to see if they are equal.
• If both sides are equal after substitution, it confirms the pair is a solution. If not, it isn't.

Let's use this method on the linear equation \(y = 2x + 6\):
If we substitute \((0, 6)\), we find \(6 = 6\), confirming it's a true solution. When substituting \((-3, 0)\), we also observe equality, so it's another solution. But, substituting \((2, -2)\) results in \(-2 eq 10\), proving it's not a solution. Always verify solutions carefully to ensure accuracy in problem-solving.

Exploring the Substitution Method

The substitution method is not just a verification tool but also essential in solving equations, especially systems of equations.
Here's how it works for verifying solutions of linear equations:

• Replace the variables in the equation with values from the ordered pair.
• Calculate the resulting expressions to determine if the equation holds true.

In the example \(y = 2x + 6\), applying substitution methodically helps confirm whether the ordered pairs like \((0, 6)\) and \((-3, 0)\) are solutions. By substituting these values into the equation, simplifications reveal true identities as balanced equations. The same method highlights when a pair is not a solution, as seen with \((2, -2)\).
The substitution method is invaluable in both simpler and complex problem-solving scenarios. It serves as a foundation in algebra, providing a consistent approach for validating or solving equations.