Problem 42 - Introductory and Intermediate Algebra for College Students 4th

Question

Determine whether each ordered pair is a solution of the given equation. $$y=8-4 x \quad(8,0),(16,-2),(3,-4)$$

Step-by-Step Solution

Answer

The pairs (8, 0) and (3, -4) are solutions to the equation, while the pair (16, -2) is not.

Step 1

Substitute the first ordered pair (8, 0) into the equation. If \(0 = 8 - 4*8\), then the pair is a solution.

Step 2

Substitute the second ordered pair (16, -2) into the equation. If \(-2 = 8 - 4*16\), then the pair is a solution.

Step 3

Substitute the third ordered pair (3, -4) into the equation. If \(-4 = 8 - 4*3\), then the pair is a solution.

Key Concepts

Linear Equations

Linear equations are one of the most fundamental concepts in algebra. They establish a relationship between two variables, often designated as \( x \) and \( y \). These equations are called "linear" because they create a straight line when graphed on a coordinate plane. A linear equation in two variables can typically be written in the standard form of \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. In the given exercise, the equation is \( y = 8 - 4x \). Here, -4 is the slope which tells us how steep the line is, and 8 is the y-intercept where the line crosses the y-axis.Characteristics of Linear Equations:

• Graphically represent a straight line.
• Consist of two variables with constant coefficients.
• Have a deterministic solution path, meaning for each \( x \), there is one specific \( y \).

This simplicity makes them easy to handle and solve, being a crucial foundation for more advanced algebraic concepts.

Ordered Pairs

Ordered pairs are essential in understanding the solutions to linear equations. They consist of two numbers arranged in a specific order, typically written as \((x, y)\). These pairs represent a point on the Cartesian coordinate system, where the first number indicates the x-coordinate and the second number indicates the y-coordinate.When we talk about a pair being a solution to an equation, we mean that substituting these values into the equation satisfies it. For example, to check if the pair \((8, 0)\) is a solution to the equation \( y = 8 - 4x \), you substitute 8 for \( x \) and 0 for \( y \). If the equation holds true, then the pair is indeed a valid solution.Key Points about Ordered Pairs:

• Always presented as \((x, y)\) with a strict order.
• Directly used in the substitution method to verify equation solutions.
• Each pair represents a unique location on a graph.

This makes ordered pairs critical in identifying solutions and graphing lines.

Substitution Method

The substitution method is a powerful technique used to determine if an ordered pair is a solution to a given equation. It involves replacing the variables in the equation with the numbers from the ordered pair to see if the resulting statement is true.Here's how it works step-by-step:

• Take the ordered pair, such as \((8, 0)\), and substitute \( x \) with 8 and \( y \) with 0 in the equation \( y = 8 - 4x \).
• Simplify the equation. For example, substitute and calculate to see if \( 0 = 8 - 4 \times 8 \).
• If what remains is a true statement, then the ordered pair is a solution to the equation. If not, it is not a solution.

This method helps to confirm whether the ordered pairs lie on the line represented by the equation. Simple logical deductions from this can be vastly beneficial for students, solidifying their understanding of how an equation behaves in relation to its graphical representation.