Problem 19 - Intermediate Algebra

Question

Determine whether the ordered pair is a solution of the system of equations. See Example 1. $$ (-0.2,0.5) ;\left\\{\begin{array}{l} 2 x+5 y=2.1 \\ 5 x+y=-0.5 \end{array}\right. $$

Step-by-Step Solution

Answer

Yes, the ordered pair (-0.2, 0.5) is a solution to the system of equations.

Step 1: Understand the Given Ordered Pair and System

We have the ordered pair \((-0.2, 0.5)\) and the system of linear equations:\[\begin{align*}2x + 5y &= 2.1 \5x + y &= -0.5\end{align*}\]We need to substitute \(x = -0.2\) and \(y = 0.5\) into each equation to see if both are satisfied.

Step 2: Substitute the Ordered Pair into the First Equation

Substitute \(x = -0.2\) and \(y = 0.5\) into the first equation:\[2(-0.2) + 5(0.5) = 2.1\]Calculate:\[-0.4 + 2.5 = 2.1\]This simplifies to:\[2.1 = 2.1\]The first equation is satisfied.

Step 3: Substitute the Ordered Pair into the Second Equation

Substitute \(x = -0.2\) and \(y = 0.5\) into the second equation:\[5(-0.2) + 0.5 = -0.5\]Calculate:\[-1 + 0.5 = -0.5\]This simplifies to:\[-0.5 = -0.5\]The second equation is also satisfied.

Step 4: Conclusion

Since both equations of the system are satisfied by the ordered pair \((-0.2, 0.5)\), it means that the pair is a solution to the system of equations.

Key Concepts

Ordered Pairs

When working with systems of linear equations, it's essential to understand the role of ordered pairs. An ordered pair, written as \(x, y\), represents a pair of numerical values assigned to variables. These pairs are used in coordinate systems to locate points and are essential for solving equations.
In systems of linear equations, you are often tasked with finding which ordered pairs satisfy both equations in the system simultaneously.

• The first number in the ordered pair represents the value of the variable \(x\).
• The second number represents the value of the variable \(y\).

In our example, the ordered pair is \(-0.2, 0.5\). We substitute these values into the equations to determine if they make both equations true. Understanding this concept is crucial as it forms the foundation for solving and verifying solutions to systems of linear equations.

Substitution Method

The substitution method is a technique used to find solutions to systems of equations. This involves substituting the values from an ordered pair directly into each equation to verify if they satisfy the conditions of the equation.
To apply the substitution method effectively:

• Start by replacing \(x\) and \(y\) in each equation with their corresponding values from the ordered pair.
• Perform the necessary arithmetic operations to check if the equations hold true.

In the example provided, we check each equation one at a time:

• For the first equation, substitute \(x = -0.2\) and \(y = 0.5\), which results in \(-0.4 + 2.5 = 2.1\). The equality holds true.
• For the second equation, substitute the same values leading to \(-1 + 0.5 = -0.5\), which also holds true.

The substitution method confirms whether the ordered pair is indeed a solution to the system by evaluating each equation separately.

Solution Verification

After applying the substitution method, solution verification is imperative. This process ensures that the solution found is correct and consistent with the equations provided. Verification involves re-evaluating each substituted equation to confirm it produces a true statement.
Here's how to verify a solution effectively:

• Ensure that after substitution, both sides of each equation are equal.
• Review calculations to avoid arithmetic or substitution errors.

For our given ordered pair \((-0.2, 0.5)\), verification showed that both equations equaled true statements: \(2.1 = 2.1\) and \(-0.5 = -0.5\). When both equations are satisfied, the pair is confirmed as a solution to the system.
Prioritizing verification in solving systems of equations helps in building confidence in problem-solving skills, ensuring that you have found an accurate and reliable solution to the problem.