Question

USING EQUATIONS Which ordered pairs are solutions of the nonlinear system? $$ \begin{aligned} &y=\frac{1}{2} x^2-5 x+\frac{21}{2} \\ &y=-\frac{1}{2} x+\frac{13}{2} \end{aligned} $$ (A) \((1,6)\) (B) \((3,0)\) (C) \((8,2.5)\) (D) \((7,0)\)

Short answer

The ordered pair that satisfies both equations is (3,0)

Step-by-step solution

Analyze the Ordered Pairs

The ordered pairs to be tested are (1,6), (3,0), (8,2.5), and (7,0). In each ordered pair, the first number is the x-value and the second number is the y-value.

Substitute Ordered Pairs in Equation 1

Substitute the x-values of the ordered pairs in Equation 1, \( y = \frac{1}{2} x^2 - 5x + \frac{21}{2} \), one by one and determine the corresponding y-values. For example, for the ordered pair (1,6), when x=1, \( y = \frac{1}{2} * 1^2 - 5*1 + \frac{21}{2} = 2 \). Hence, (1,6) is not a solution of Equation 1.

Substitute in Equation 2

Next, substitute the x-values of the ordered pairs in Equation 2, \( y= -\frac{1}{2}x + \frac{13}{2} \), one by one and compare the derived y-values with the y-values of the ordered pairs. For the pair (1,6), substitution yields \( y = -\frac{1}{2}*1 + \frac{13}{2} = 6 \). Hence, (1,6) is a solution of Equation 2 but not of Equation 1.

Repeat for All Ordered Pairs

Repeat steps 2 and 3 for all the given ordered pairs. An ordered pair is a solution of the system if it satisfies both Equations 1 and 2.

Key concepts

Ordered Pairs

In the realm of mathematics, particularly when dealing with systems of equations, an essential concept is that of ordered pairs. These pairs consist of two elements, typically denoted as \(x, y\). The essence of these pairs is that they provide coordinates within a two-dimensional plane, where the first element represents the x-coordinate, and the second represents the y-coordinate.

• The x-value indicates a position along the horizontal axis.
• The y-value indicates a position along the vertical axis.

In the context of solving equations, these pairs serve as potential solutions that we must verify to satisfy both equations involved in the system. An ordered pair is considered a valid solution only if substituting its x and y values into each equation results in true statements.

Equations

Equations are mathematical statements that assert the equality of two expressions. When given a system of equations, especially nonlinear ones like those in this problem, we work with multiple equations simultaneously. In this exercise:

• Equation 1 is \(y = \frac{1}{2}x^2 - 5x + \frac{21}{2}\). This is a quadratic equation, meaning it represents a parabolic curve on a graph. Such equations often have more complex behavior and multiple potential solutions.
• Equation 2 is \(y = -\frac{1}{2}x + \frac{13}{2}\). This one is linear, forming a straight line when graphed. The solutions to this equation are easier to determine due to its straightforward nature.

By substituting the x-value from each ordered pair into these equations, and by verifying if the results match the corresponding y-values, we confirm if the pair satisfies the equation. Both equations need to be true for an ordered pair to be a solution of the nonlinear system.

Solution Verification

Solution verification is a critical step in solving systems of equations. This process involves checking whether the ordered pairs actually satisfy the given equations. It's like a detective work, confirming if the potential solutions truly hold up against both equations at the same time.

Firstly, for Equation 1, substituting the x-value of an ordered pair will yield a particular y-value using the equation's formula. For instance, substituting \(x = 1\) in Equation 1 gives a result for y. This result must be compared with the y-value in the ordered pair. If they match, the pair satisfies Equation 1.

Secondly, apply the same method for Equation 2. For each ordered pair, ensure that substituting their x-values and comparing derived y-values with given ones results in equality. Only if both equations verify true for an ordered pair, can it be declared a genuine solution of the nonlinear system.

• Order matters: always check both equations independently.
• A true solution makes both equations hold true simultaneously.

This thorough checking ensures accuracy and verifies the mathematical integrity of the solutions.