Problem 16 - Algebra 1

Question

Decide whether the given ordered pair is a solution of the equation. \(-5 x-8 y=15,(-3,0)\)

Step-by-Step Solution

Answer

Yes, the ordered pair \((-3,0)\) is a solution to the equation \(-5x-8y=15\).

Step 1: Identify Ordered Pair

The ordered pair given is (-3,0), in which -3 is the x-value and 0 is the y-value.

Step 2: Substitute x and y into Equation

Substitute the values of x and y from the ordered pair into the given equation, which will look like this: \(-5(-3)-8(0)=15\)

Step 3: Simplify the Equation

Now, we simplify the equation. The result will be 15 on the left-hand side and 15 on the right-hand side: \(15 = 15\)

Step 4: Compare both sides of the equation

The left-hand side equals the right-hand side, so the ordered pair \((-3,0)\) is indeed a solution to the equation.

Key Concepts

Understanding Ordered Pairs

An ordered pair is a fundamental concept in mathematics, often used in linear equations. An ordered pair is represented as

• (x, y), where 'x' is the first element typically representing the horizontal position on a graph,
• 'y' is the second element representing the vertical position.

Understanding this notation helps in identifying solutions for equations. When dealing with linear equations, ordered pairs indicate potential solutions that can satisfy the equation when substituted.
Consider each part of the ordered pair as coordinates that may potentially lie on the graph of the equation.

The Substitution Method

The substitution method is a reliable way to verify if an ordered pair is a solution to an equation. This method involves replacing the variables in the equation with the corresponding values from the ordered pair. For instance:

• Take the ordered pair (-3, 0) from the equation \(-5x - 8y = 15\).
• Substitute \(x = -3\) and \(y = 0\) into the equation, leading to \(-5(-3) - 8(0)\).

Performing substitution helps convert the equation into a simpler numerical expression that is easier to evaluate. By plugging in the x and y values, you can ascertain whether the equation holds true for the ordered pair. This verification hinges on whether both sides of the equation yield the same result after substitution.

Solution Verification

Solution verification involves confirming that an ordered pair satisfies a given equation. This is achieved by comparing the simplified expression after substitution to the original equation's right side.

As shown:

• Once we substitute \(x = -3\) and \(y = 0\) into \(-5x - 8y = 15\), it simplifies to \(15 = 15\).
• If the left side equals the right side, then the ordered pair is indeed a solution.

This comparison step is crucial, as it verifies the correctness of the evaluation. If the expression holds true (i.e., both sides are equal), the ordered pair is a solution to the equation. If not, it implies errors in calculation or indicates that the pair does not satisfy the equation.