Problem 15 - Algebra 1

Question

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

Step-by-Step Solution

Answer

The ordered pair (-2,8) is not a solution to the equation 2y-4x=8.

Step 1: Substitute the Ordered Pair into the Equation

Substitute the pair (-2,8) into the equation in place of x and y. Thus, the equation becomes \(2*8 - 4*(-2) = 8\). Recall that the first part of the pair replaces x, and the second part replaces y.

Step 2: Simplify the Equation

Simplify the equation. This results in \(16 + 8 = 8\).

Step 3: Check If the Equation Is True

Check if the simplified expression equals the number on the other side of the equation. In this case, \(24 \not= 8\), and thus it's not a valid expression.

Key Concepts

Algebraic Equations

Algebraic equations are mathematical statements in which expressions containing variables are set equal to each other. These variables represent unknown values that we seek to determine. In the given exercise, the equation is written as \(2y - 4x = 8\). This means that when we replace the variables \(x\) and \(y\) with specific numbers, the resulting expression on the left side should equal 8, which is the number on the right side of the equation. Understanding how to manipulate and solve algebraic equations is critical for verifying if certain pairs, known as ordered pairs, satisfy the equation. An ordered pair, such as \((-2,8)\), signifies a specific solution where \(x = -2\) and \(y = 8\). The main goal with such exercises is to clarify whether substituting these values for \(x\) and \(y\) in the equation will keep the equation balanced, meaning both sides of the equation are equal. Learning algebraic equations also lays the groundwork for more advanced mathematical concepts, helping improve problem-solving skills and analytical thinking.

Solution Verification

In the context of algebra, solution verification is the step where we check if a given ordered pair satisfies an equation. Verification ensures that the mathematical expression remains true when specific values replace variables. To verify the solution of the equation \(2y - 4x = 8\) with the ordered pair \((-2,8)\), you follow these simple steps:

• Substitute \(x = -2\) and \(y = 8\) into the equation.
• Calculate the left-hand side of the equation.
• Compare the result with the right-hand side, which is 8.

This process allows you to see if both sides match. In this problem, after substitution and calculation, the left-hand side results in 24. However, since 24 does not equal 8, the ordered pair \((-2, 8)\) is not a solution. Verification is key to ensuring that substituted values truly satisfy the equation before considering them as the solution. This is a crucial skill for checking work in mathematics and related fields.

Substitution Method

Using the substitution method helps employ a systematic approach to verify solutions to algebraic equations. This involves directly replacing variables with numbers from an ordered pair into the equation. It is a crucial method for checking if an ordered pair is a solution to a given equation. Here's how you can effectively use the substitution method:

• Identify which values correspond to \(x\) and \(y\) in the ordered pair.
• Replace \(x\) and \(y\) in the equation with these values directly.
• Perform arithmetic operations to simplify the equation.

For example, using the pair \((-2, 8)\) in \(2y - 4x = 8\), you substitute \(x\) with -2 and \(y\) with 8, resulting in \(2(8) - 4(-2) = 8\). Calculating further, \(16 + 8 = 24\), which does not equal 8, showing the pair is not the solution. The substitution method not only helps verify solutions but is also a fundamental technique in solving systems of equations where you apply similar steps to find missing variable values. It makes working through complex equations more straightforward and increases mathematical accuracy.