Problem 38 - Introductory and Intermediate Algebra for College Students 4th

Question

Determine whether each ordered pair is a solution of the given equation. $$y=4 x \quad(3,12),(12,3),(-5,-20)$$

Step-by-Step Solution

Answer

The ordered pairs (3,12) and (-5,-20) are solutions to the equation \(y=4x\), while the ordered pair (12,3) is not a solution.

Step 1: Understanding the Ordered Pair

In an ordered pair, the first value refers to 'x' and the second value refers to 'y'. So for each ordered pair, 'x' values are 3, 12, and -5 while 'y' values are 12, 3, and -20 respectively.

Step 2: Substituting the Values into the Equation

The given equation is \(y=4x\). We substitute each ordered pair into this equation. For the pair (3,12), we substitute 'x' with 3 and 'y' with 12 to see if \(12 = 4*3\). For the pair (12,3), we substitute 'x' with 12 and 'y' with 3 to see if \(3 = 4*12\). For the pair (-5,-20), we substitute 'x' with -5 and 'y' with -20 to see if \(-20 = 4*(-5)\).

Step 3: Verifying Each Pair

Now we check if the equation holds for each ordered pair. For (3,12), we find that \(12=4*3\) (12=12), so it holds true. For (12,3), we find that \(3!=4*12\) (3!=48), so it doesn't hold true. For (-5,-20), we find that \(-20 = 4*(-5)\) (-20=-20), so it holds true.

Key Concepts

Linear Equations

A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. Linear equations can be written in the form of ax + by = c, where a, b, and c are constants, and x and y are variables. These equations graph as straight lines when plotted on a coordinate system, hence the term 'linear'.

One of the simplest forms of a linear equation is when it is solved for one variable in terms of the other, usually y in terms of x, which looks like y = mx + b where m represents the slope of the line, and b is the y-intercept. In the exercise example, y = 4x, the coefficient 4 is the slope, meaning for each unit increase in x, y increases by 4 times that amount. There is no y-intercept present, which means the line passes through the origin (0,0).

Solving linear equations often involves finding values for the variables that make the equation true. These values are called solutions to the equation. In the context of our example, solutions are ordered pairs (x, y) which satisfy the equation, i.e., if you plug them into the equation, both sides will be equal.

Substitution Method

The substitution method is an algebraic technique used to find the exact solution to a system of linear equations. Essentially, it involves replacing one variable with another to reduce the equations to a single variable form. This simplifying process can make equations more manageable and easier to solve.

In the context of the given problem, we apply substitution by plugging in the x and y values from each ordered pair into the equation y = 4x. If the equation balances, meaning both sides equal the same number, then the ordered pair is a solution to the linear equation. Here's a step-by-step look at how this is done:

• Take the ordered pair (3,12).
• Substitute 3 for x and 12 for y in the equation.
• Check if 12 = 4 * 3.
• If true, (3,12) is a solution to the equation; if false, it is not.

This method not only helps in finding solutions to single linear equations but also lays the groundwork for solving more complex systems of equations in algebra.

Algebraic Reasoning

Algebraic reasoning is the process of using mathematical reasoning to solve algebraic problems. It goes beyond memorizing procedures and formulas, requiring the ability to understand the relationships between variables, make generalizations, and formulate and justify arguments based on the structure of the algebraic system.

In the exercise provided, algebraic reasoning is at work when we verify whether the ordered pairs are solutions to the equation. After substituting the values, logic and reasoning allow us to confirm the truth or falsity of our equations. For instance, under scrutiny, we realize that while the pair (3,12) adheres to the equation (since 12 is indeed four times 3), the pair (12,3) does not because 3 is not four times 12; the product of 4 and 12 exceeds 3 significantly, demonstrating a clear misunderstanding of the linear relationship. This aspect of reasoning is crucial for understanding algebra at a deeper level which allows students to solve problems effectively and develop mathematical intuition.