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Chapter 8: Problem 3

Is a system of linear equations with no solution consistent or inconsistent?

Short Answer

Expert verified
A system of linear equations with no solution is inconsistent.

Step by step solution

01

Understanding Systems of Linear Equations

A system of linear equations is a set of two or more linear equations that have the same variables. These systems can be graphically interpreted as lines on a plane. The solutions to these systems are the coordinates where the lines intersect.
02

Defining Consistent and Inconsistent Systems

A system of linear equations is termed as 'consistent' if it has at least one solution, and 'inconsistent' if it has no solutions. This is primarily determined by how the lines intersect: if they intersect at at least one point (which can be a unique solution or infinitely many if the lines coincide), the system is consistent; if they never intersect (i.e., the lines are parallel), the system is inconsistent.
03

Answering the exercise question

With these definitions in mind, we know that a 'system of linear equations with no solution' does not have any points of intersection. Therefore, it is an inconsistent system.

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Key Concepts

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

Consistent System
When we talk about a consistent system in the context of linear equations, we are referring to a set of equations that share at least one common solution.
In simple terms, if you can find a point (x,y) that satisfies all the equations in the system simultaneously, then the system is consistent. This can occur in two ways:
  • **Unique Solution:** If two lines intersect at a single point, the system has one unique solution. This means there's exactly one set of values for the variables that satisfies both equations.
  • **Infinite Solutions:** If the lines overlap completely, meaning they are the same line, every point on the line is a solution to both equations, resulting in infinitely many solutions.
Visualizing these scenarios on a graph, a consistent system has at least one intersection where the lines meet, confirming the shared solution(s).
Inconsistent System
A system of linear equations is called inconsistent if it has no solutions. Imagine this as trying to find a common intersection of two lines on a graph, but they never meet.
In the case of an inconsistent system:
  • **Parallel Lines:** The lines are parallel, meaning they have the same slope but different y-intercepts. Because they never intersect, there are no points in common, leading to no solution at all.
Inconsistent systems are important to identify because no matter how hard we try to solve them, we won't find a satisfactory answer where both equations hold true simultaneously. In practical terms, it tells you that there is a contradiction in your problem setup, such as asking two impossibly different conditions to hold at once.
Solutions to Systems of Equations
Solutions to systems of linear equations are special because they reveal the point(s) at which the equations come together. Determining the solution often involves finding intersections on a graph. There are three main types of solutions that can occur when dealing with linear systems:
  • **No Solution:** As discussed, when the system is inconsistent, the graphs of the equations do not meet.
  • **One Solution:** Think of this as the golden ticket in consistency. Here, the lines intersect precisely at one point. This single intersection is the solution to the entire system, solving all the equations at once.
  • **Infinitely Many Solutions:** This outcome occurs when the lines are completely on top of each other, making every point on the line a solution to both equations.
Thus, finding solutions involves observing how the equations relate to each other graphically and analytically, leading to deeper insights about the nature of the relationships they define.

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Most popular questions from this chapter

Evaluate the determinant, in which the entries are functions. Determinants of this type occur when changes of variables are made in calculus. $$\left|\begin{array}{ll} x & \ln x \\ 1 & 1 / x \end{array}\right|$$

Sales The projected sales \(S\) (in millions of dollars) of two clothing retailers from 2015 through 2020 can be modeled by \(\left\\{\begin{array}{ll}S-149.9 t=415.5 & \text { Retailer } \mathrm{A} \\\ S-183.1 t=117.3 & \text { Retailer } \mathrm{B}\end{array}\right.\) where \(t\) is the year, with \(t=5\) corresponding to 2015 (a) Solve the system of equations using the method of your choice. Explain why you chose that method. (b) Interpret the meaning of the solution in the context of the problem. (c) Interpret the meaning of the coefficient of the \(t\) -term in each model. (d) Suppose the coefficients of \(t\) were equal and the models remained the same otherwise. How would this affect your answers in parts (a) and (b)?

A florist is creating 10 centerpieces for the tables at a wedding reception. Roses cost \(\$ 2.50\) each, lilies cost \(\$ 4\) each, and irises cost \(\$ 2\) each. The customer has a budget of \(\$ 300\) for the centerpieces and wants each centerpiece to contain 12 flowers, with twice as many roses as the number of irises and lilies combined. (a) Write a linear system that represents the situation. (b) Write a matrix equation that corresponds to your system. (c) Solve your linear system using an inverse matrix. Find the number of flowers of each type that the florist can use to create the 10 centerpieces.

The sums have been evaluated. Solve the given system for \(a\) and \(b\) to find the least squares regression line for the points. Use a graphing utility to confirm the results. $$\left\\{\begin{array}{l} 5 b+10 a=20.2 \\ 10 b+30 a=50.1 \end{array}\right.$$

Determine whether the statement is true or false. Justify your answer. If a square matrix has an entire row of zeros, then the determinant will always be zero.
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