Problem 62 - Intermediate Algebra

Question

Decide whether the ordered pair \((-4,3)\) is a solution of the system of linear equations. $$ \left\\{\begin{array}{l} y=2 x+11 \\ \frac{x}{2}+y=0 \end{array}\right. $$

Step-by-Step Solution

Answer

The ordered pair \((-4, 3)\) is not a solution to the system of equations.

Step 1: Substitute into First Equation

The first equation is given as \( y = 2x + 11 \). Substitute \( x = -4 \) and \( y = 3 \) into this equation. Check if both sides of the equation are equal. Substitute: \( 3 = 2(-4) + 11 \). Calculate the right side: \( 3 = -8 + 11 \), and simplify to get \( 3 = 3 \). Since both sides are equal, \((-4, 3)\) satisfies the first equation.

Step 2: Substitute into Second Equation

The second equation is \( \frac{x}{2} + y = 0 \). Substitute \( x = -4 \) and \( y = 3 \). Check if both sides are equal.Substitute: \( \frac{-4}{2} + 3 = 0 \). Calculate the left side: \( -2 + 3 = 1 \). This results in \( 1 = 0 \), which is not true. Therefore, \((-4, 3)\) does not satisfy the second equation.

Step 3: Conclusion on the Ordered Pair

Since the ordered pair \((-4, 3)\) satisfies the first equation but not the second equation, it is not a solution to the system of equations.

Key Concepts

Ordered Pairs

In mathematics, an ordered pair is simply two numbers enclosed in parentheses, such as \((-4, 3)\). This pair represents a point on a coordinate plane, where the first number is the x-value and the second is the y-value. Ordered pairs are crucial for graphing and solving systems of equations. Each part of an ordered pair provides specific information about a point's position relative to the x-axis and the y-axis. When we talk about systems of linear equations, an ordered pair may represent a potential solution. This is because it can satisfy both equations in the system simultaneously. Recognizing that ordered pairs are tested by plugging into each equation is fundamental when determining if they solve a given system.
In our specific exercise, we wanted to see if \((-4, 3)\) solves a system of linear equations. This involved substituting x and y into both equations and checking if they held true.

Solution Verification

Solution verification is the process of checking whether a proposed solution, like an ordered pair, satisfies a system of equations. To verify a solution, substitute the variables in each equation with the values in the ordered pair. You need both to result in true statements for the solution to be valid.
In the given problem, we worked through a two-step verification process:

• First, we substituted the values \((-4, 3)\) into the first equation, which resulted in a true statement, verifying the ordered pair for this equation.
• Next, we substituted those same values into the second equation. However, this did not result in a true statement, showing it did not satisfy the second equation.

By identifying where the solution failed, we concluded that the ordered pair was not a solution to the complete system.

Substitution Method

The substitution method is a common approach used to solve systems of linear equations. It involves solving one of the equations for one variable in terms of the other and using this expression to replace the variable in the second equation. By doing so, we reduce the system to a single equation with one variable, making it easier to solve.
However, in this exercise, our focus was on substitution for verification rather than for solving. We substituted the x-value and y-value from the ordered pair into each equation of the system. This step confirms whether the values are solutions by checking if the equations become true statements.

• This method allowed us to effectively test the first equation and verify it was satisfied by \((-4, 3)\).
• Although the substitution in the second equation did not confirm the pair as a solution, it clearly showcased the system's limitations with these particular values.

Utilizing substitution, whether for solving or verifying, is a powerful tool in managing equations.