Problem 109 - Introductory and Intermediate Algebra for College Students 4th

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The ordered pair \((3,4)\) satisfies the equation $$2 y-3 x=-6$$

Step-by-Step Solution

Answer

The statement is false, given \(-1 ≠ -6\). To make it a true statement, we need to find an ordered pair \((x, y)\) that when substituted into the equation \(2y - 3x\), equals to \(-6\).

Step 1: Substitute the given ordered pair into the equation

Replace \(x\) with 3 and \(y\) with 4 in the equation \(2y -3x = -6\). After doing so, the equation becomes \(2*4 - 3*3 = -6\)

Step 2: Simplify the equation

Perform the multiplications first (since they are priority according to the order of operations or BIDMAS/BODMAS rule), resulting in the equation \(8 - 9 = -6\)

Step 3: Further simplify the equation

Subtract 9 from 8, which gives -1. Thus, the equation simplifies to \(-1 = -6\)

Step 4: Verify the equality

Check if both sides of the equation are equal. If \(-1\) is equal to \(-6\), then the statement is true; otherwise, it is false. In this case, \(-1 ≠ -6\), hence the statement is false

Key Concepts

Ordered Pairs

In mathematics, an ordered pair is a fundamental concept used to describe an exact location or point on a plane. An ordered pair is written in the form \((x, y)\), where \(x\) represents the horizontal coordinate, and \(y\) represents the vertical coordinate. These coordinates specify a unique position on a grid, similar to a map.Ordered pairs are essential in many mathematical areas such as graphing equations, geometry, and algebra. When working with equations, ordered pairs are often used to indicate specific solutions or points that satisfy the equation. For example, if an equation has multiple solutions, each can be represented as an ordered pair.To better understand, always consider the order: the first element \(x\) is a different component from the second element \(y\). This distinction is crucial, as switching them changes their significance. Practicing with ordered pairs helps solidify their role in solving algebraic problems and verifying specific solutions.

Equation Solving

Equation solving is the process of finding values for variables that make an equation true. In algebra, equations are fundamental, as they represent a form of mathematical statement where both sides must be equal. To solve an equation means to determine the value of the variable(s) involved, which satisfies the given condition. This often requires performing a series of operations such as addition, subtraction, multiplication, or division on both sides of the equation. Simplifying the terms and systematically isolating the variable results in the solution. For example, in the problem provided, you substitute the numbers from the ordered pair into the equation and check if the sides' equality holds. Equation solving is crucial for various real-world applications, from calculating simple daily expenses to solving complex scientific problems.

BODMAS Rule

The BODMAS rule is a helpful guideline in mathematics that outlines the sequence of operations to perform in order to accurately solve an expression. BODMAS stands for:

• Brackets
• Orders (i.e., powers and roots)
• Division and Multiplication (from left to right)
• Addition and Subtraction (from left to right)

The correct application of the BODMAS rule is crucial in simplifying mathematical expressions correctly. For example, in our step-by-step solution, multiplications were performed before subtraction, following the BODMAS principle. This ensures that everyone arrives at the same result regardless of the complexity of the expression. Using BODMAS prevents errors and simplifies the process of evaluating mathematical expressions. It's essential always to perform calculations in this order to derive correct results.

Verification of Solutions

Verification of solutions involves checking if a proposed solution actually satisfies the given equation. This is a crucial step in the mathematical problem-solving process, ensuring that the right conclusions are drawn from the calculation.To verify a solution, substitute the values back into the original equation. After substitution, simplify as per the BODMAS rule, and check if both sides of the equation are indeed equal. If they are, the solution is valid; if not, the proposed solution is incorrect.In our example, substituting the ordered pair \((3, 4)\) into the equation results in \(-1 = -6\), which is untrue. This tells us that the ordered pair does not satisfy the equation, making the statement false.Verification helps confirm our solutions and ensures that all calculations align with the intended mathematical requirements. Practicing verification can reinforce understanding and accuracy in solving algebraic problems.