Problem 20 - College Algebra with Modeling and Visualization

Question

Determine which ordered pairs are solutions to the given system of equations. State whether the system is linear or nonlinear. $$ \begin{aligned} &\begin{array}{l} (3,2),(3,-4),(5,0) \\ x-y=5 \end{array}\\\ &2 x+y=10 \end{aligned} $$

Step-by-Step Solution

Answer

(5, 0) is a solution; the system is linear.

Step 1: Identify the System Type

The system consists of the equations \( x-y=5 \) and \( 2x+y=10 \). Both equations are of the first degree, as the highest power of the variables is 1. Thus, this system is linear.

Step 2: Check the Ordered Pair (3, 2)

Substitute \( x = 3 \) and \( y = 2 \) into the first equation: \( 3 - 2 = 1 \), which is not equal to 5. The first equation is not satisfied, so this pair is not a solution.

Step 3: Check the Ordered Pair (3, -4)

Substitute \( x = 3 \) and \( y = -4 \) into the first equation: \( 3 - (-4) = 3 + 4 = 7 \), which is not equal to 5. This pair does not satisfy the first equation, so it is not a solution.

Step 4: Check the Ordered Pair (5, 0)

Substitute \( x = 5 \) and \( y = 0 \) into the first equation: \( 5 - 0 = 5 \), which satisfies the equation. Now substitute into the second equation: \( 2(5) + 0 = 10 \), which satisfies the second equation. This pair satisfies both equations, so it is a solution.

Key Concepts

System of Equations

A system of equations consists of two or more equations that share the same set of unknowns. The purpose of solving a system of equations is to find the values of the unknowns that satisfy all given equations simultaneously. In the example provided, we have:

• Equation 1: \( x - y = 5 \)
• Equation 2: \( 2x + y = 10 \)

These equations involve two variables, \( x \) and \( y \). Solving this system means finding values for \( x \) and \( y \) that make both equations true at the same time. It's like finding a point where two lines intersect on a graph. Then, we can test various ordered pairs to see which ones solve the entire system of equations.

Ordered Pairs

An ordered pair is a pair of numbers used to locate a point on a coordinate plane. It is usually written in the form \((x, y)\). In the context of a system of equations, ordered pairs represent potential solutions, where \( x \) and \( y \) are the respective values.
For example, in the exercise given, the ordered pairs \((3, 2)\), \((3, -4)\), and \((5, 0)\) were tested:

• \((3, 2)\) does not satisfy the first equation.
• \((3, -4)\) also fails the first equation.
• \((5, 0)\) satisfies both equations, making it a solution to the system.

We test each ordered pair by substituting the values into the equations to see if the equation holds true. Where both equations are satisfied, the ordered pair is a solution to the system.

Linear vs Nonlinear

Understanding the difference between linear and nonlinear systems is crucial. A linear system is made up of linear equations, where each term is either a constant or the product of a constant and a single variable. Linear equations graph as straight lines in a two-dimensional plane.
In this scenario, both the equations \( x - y = 5 \) and \( 2x + y = 10 \)are linear, meaning they graph as straight lines and have the standard form \( ax + by = c \). Each term’s variable has an exponent of one. Linear systems generally have one unique solution, infinitely many solutions, or no solutions at all, depending on how the lines interact (e.g., intersecting, parallel, or coincident).
Nonlinear systems, on the other hand, involve equations of higher degrees, such as quadratic equations, and graph as curves like parabolas. Such systems can have more complex solution sets.

Solution of Equations

The solution to a system of equations is the set of all ordered pairs that satisfy all the equations in the system. Finding solutions involves testing different pairs or using algebraic methods to determine valid points.
In our specific example, we verified potential solutions by substituting each ordered pair into the equations:

• The ordered pair \((3, 2)\) did not satisfy the first equation.
• The ordered pair \((3, -4)\) did not meet the first equation’s requirement either.
• Ultimately, the pair \((5, 0)\) met both conditions, making it a valid solution.

Solutions can also be assessed using graphical or algebraic techniques such as substitution or elimination, where one solves for one variable in terms of another to find intersecting points.