Problem 18 - Algebra 1: Concepts and Skills

Question

Determine whether the ordered pair is a solution of the equation. $$ 2 y-4 x=8,(-2,8) $$

Step-by-Step Solution

Answer

No, the ordered pair (-2,8) is not a solution to the equation \(2y - 4x = 8\).

Step 1: Identify the ordered pair

An ordered pair is simply two numbers written in a certain order. In this case, the ordered pair is (-2,8). Normally, these pairs are written as (x,y), so that -2 is x and 8 is y.

Step 2: Substitute the values

Substitute x=-2 and y=8 into the given equation, yielding \(2*8 - 4*(-2) = 16 + 8 = 24\) on the left-hand side.

Step 3: Compare the result with the right-hand side

Next, compare 24 (the result on the left-hand side) with 8 (the right-hand side of the equation). Since these are not equal (24 ≠ 8), the given ordered pair is not a solution to the equation.

Key Concepts

Solving Equations

To determine whether an ordered pair is a solution of an equation, we must substitute the pair's values into the equation and verify if the equality holds true. This process involves:

• Identifying the variables and constants in an algebraic equation.
• Ensuring each side of the equation is balanced after substitution.

For the specific equation provided, \( 2y - 4x = 8 \), we have to plug in the values from the ordered pair, where \( x = -2 \) and \( y = 8 \). Therefore, the left side becomes \( 2(8) - 4(-2) = 16 + 8 = 24 \). If both sides match after such substitution, the pair is considered a solution. In this case, however, the resulting value of 24 does not equal the right side of the equation, which is 8, thus confirming the ordered pair is not a solution.
Solving equations in this manner relies on maintaining the equality by accurately substituting and simplifying.

Systems of Equations

Systems of equations involve multiple equations working together to determine the values of variables. Generally, solving these systems requires finding values that satisfy all of the included equations simultaneously. This provides a unique solution, multiple solutions, or no solution. In a two-variable system, we often deal with forms like:

• Linear equations such as \( ax + by = c \)
• Quadratic or other polynomial equations

While the original exercise deals with a single equation, gaining an understanding of systems is crucial for broader mathematical problem-solving. When dealing with systems, one common method to solve them is to find ordered pairs where all the individual equations hold true. This particular exercise emphasizes the first step in such systems—checking if any given pair is a solution to one of the equations in the system.

Substitution Method

The substitution method is one of the ways to solve systems of equations, as it allows for solving one equation for one variable, and then substituting this expression into the other equation. Even though our initial exercise does not require solving a system of equations, understanding substitution can dramatically simplify finding solutions:

• Solve one equation for one of the variables.
• Substitute this expression into the other equation.
• Solve for the remaining variable.
• Back-substitute to find the value of the first variable.

When implemented in a system where one equation is already simplified for a variable, the substitution method can quickly lead to a solution. It's an indispensable tool in algebra, converting a system to simpler equations, eventually revealing the relationships between variables.