Problem 50 - Algebra 1: Concepts and Skills

Question

Determine whether the ordered pair is a solution of the equation. (Lesson 4.2) $$ 3 x-6 y=-2,(-4,-2) $$

Step-by-Step Solution

Answer

No, (-4, -2) is not a solution to the equation \(3x - 6y = -2\).

Step 1: Identify the ordered pair and equation

The ordered pair given is (-4, -2) and the equation is \(3x - 6y = -2\). We need to substitute x with -4 and y with -2 in the equation and see if both sides become equal.

Step 2: Substituting the values

If we substitute x with -4 and y with -2, the equation becomes \(3(-4) - 6(-2) = -2\), which simplifies to \(-12 + 12 = -2\).

Step 3: Verification

By simplifying the equation, we obtain \(0 = -2\). This signifies that the left and the right sides of the equation are not equal, meaning the ordered pair (-4, -2) is not the solution to the equation \(3x - 6y = -2\)

Key Concepts

Understanding Ordered Pairs

In algebra, an ordered pair is a set of numbers written in a specific pattern or order, usually denoted as \(x, y\). The first value typically represents the x-coordinate, which tells you how far the point is along the horizontal axis, while the second value represents the y-coordinate, indicating how far the point is on the vertical axis.
For example, in the ordered pair (-4, -2), -4 is the x-coordinate and -2 is the y-coordinate. When you're told to find if an ordered pair is a solution to an equation, your task is to see if these coordinates satisfy the given algebraic equation. This simply means plugging the values of x and y into the equation to see if it holds true. If it does, the point is on the line represented by the equation; otherwise, it's not. This concept is fundamental in graphing and understanding linear relationships.

Mastering the Substitution Method

The substitution method is a powerful tool in algebra for evaluating equations. This method involves substituting the given values of variables directly into the equation. Let's break it down with our example equation: \(3x - 6y = -2\) and the ordered pair (-4, -2).
Here's how it works:

• Take the first value of the ordered pair, which represents x. Substitute -4 wherever x appears in the equation.
• Next, substitute -2 for every instance of y in the equation.
• So the equation becomes \(3(-4) - 6(-2)\).

This substitution changes the original algebraic equation into a simple arithmetic problem, which you can then solve to check if both sides are equal. If they are, then the ordered pair is a solution to the equation. If not, as in our exercise where we end up with \(0 = -2\), it's not.

Solution Verification: Checking Your Work

After substituting the values and simplifying the equation, the crucial final step is solution verification. Solution verification confirms whether the ordered pair truly satisfies the equation.
In our exercise, after substitution, we simplified the equation to get \(-12 + 12 = -2\). When simplified further, this results in \(0 = -2\), which clearly indicates that both sides of the equation are not equal. Hence, the ordered pair (-4, -2) is not a solution.
Verification not only checks our work for accuracy but also helps solidify the understanding of the relationship between the ordered pair and the equation. By always confirming the solution, you ensure mathematical precision and reinforce algebraic concepts.