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Chapter 5: Problem 92

Consider a system of linear equations in two variables in which the solution set is \(\\{(x, y) \mid y=x+2\\}\). Why do we say that the equations in the system are dependent?

Short Answer

Expert verified
The equations are dependent because they describe the same line and have the same solution set.

Step by step solution

01

Identify the Solution Set

The solution set provided is \( \{ (x, y) \mid y = x + 2 \} \). This means that any point (x, y) that satisfies the equation y = x + 2 is a solution to the system.
02

Interpret the Given Information

The equation y = x + 2 represents a line in the Cartesian plane. The given solution set indicates that all solutions lie on this line.
03

Define Dependent Equations

Two or more equations in a system are dependent if they describe the same line. This means one equation can be derived from the other by algebraic manipulation.
04

Analyze the Relationship Between Equations

Since the solution set contains all the points on the line y = x + 2, any system of equations with this unique solution set must consist of equations that represent the same line, making them dependent.
05

Conclude Dependence

Because the equations in the system have the same set of solutions and thus describe the same line, they are dependent.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

linear equations
Linear equations are equations that form a straight line when plotted on the Cartesian plane. They are usually in the form of! y = mx + b, where m represents the slope and b represents the y-intercept. For example, in the exercise provided, the linear equation is y = x + 2. This means that for every unit increase in x, y increases by the same amount, creating a straight line with a slope of 1 and a y-intercept of 2.
solution set
A solution set is a collection of all possible solutions to a given equation or system of equations. In our exercise, the solution set is \[\{(x, y) \mid y = x + 2\}\]. This tells us that any point (x, y) that lies on the line defined by \[y = x + 2\] is a solution to the system. This is crucial in understanding dependent equations, as the solution set provides insight into whether the equations describe the same line.
Cartesian plane
The Cartesian plane is a two-dimensional plane defined by a horizontal axis (x-axis) and a vertical axis (y-axis). When plotting linear equations, each pair (x, y) corresponds to a point on this plane. In our exercise, the line \[y = x + 2\] can be plotted on the Cartesian plane, forming a straight line that intersects the y-axis at 2. This visual representation helps in understanding the relationships between different linear equations.
algebraic manipulation
Algebraic manipulation involves using algebraic techniques to simplify or rearrange equations. In the context of our exercise, we use algebraic manipulation to show that two equations are dependent. For instance, if we have another equation such as 2y = 2x + 4, we can simplify it to y = x + 2 by dividing both sides by 2. This shows that the new equation describes the same line as the first, indicating that they are dependent.
system of equations
A system of equations consists of multiple equations that are considered simultaneously. They can have a single solution, many solutions, or no solution at all, depending on whether they are consistent, inconsistent, or dependent. In our exercise, the system is determined to be dependent because both equations describe the same line. This means that the system does not produce a unique solution indicated by multiple intersecting lines, but rather, a single line that all solutions lie on.

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