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Chapter 3: Problem 2

Classify each of the following statements as either true or false. It is possible for a system of equations to have an infinite number of solutions.

Short Answer

Expert verified
True

Step by step solution

01

Identify Types of Solutions

First, understand that a system of linear equations can have three possible types of solutions: no solution, exactly one solution, or infinitely many solutions.
02

Define Infinite Solutions

An infinite number of solutions occur when the equations in the system are dependent, meaning one equation is a multiple of the other. In other words, both equations describe the same line in a 2-dimensional plane.
03

Validate Statement

Since it is mathematically possible for a system of equations to describe the same line and thus have an infinite number of solutions, the statement is true.

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Key Concepts

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

infinite solutions
When solving systems of linear equations, it's possible to encounter a scenario where an infinite number of solutions exist. This happens when the equations describe the same line, overlapping perfectly. Imagine two equations like this: y = 2x + 3 and 2y = 4x + 6. If you simplify the second equation, you end up with y = 2x + 3, which is identical to the first equation.

  • If every point on one line is also on the other, then the system has infinitely many solutions.

An easy way to verify this is to rearrange the equations into their simplest forms and see if they match.
dependent equations
Dependent equations are central to understanding systems with infinite solutions. These are equations that rely on each other for their configurations. For instance, in a system where one equation is simply a multiple of the other, those two equations are dependent.

This means there is not new information provided by having both equations: they effectively describe the same constraint.
  • In geometrical terms, dependent equations describe the same line in the plane.
  • They overlap completely, hence any point on this line is a solution to both equations.

Dependent equations can be identified by simplifying them and checking whether one is a scaled version of the other.
linear equations
Linear equations are equations where the highest power of the variable is one. These equations form straight lines when graphed on a coordinate plane. The general form of a linear equation in two variables is: ax + by = c, where a, b, and c are constants.

Whether a system of linear equations has one, none, or infinitely many solutions depends on the relationships between the lines they describe.
  • If the lines intersect at one point, the system has a single solution.
  • If the lines are parallel, there is no solution.
  • If the lines coincide (are dependent), there are infinitely many solutions.

To solve a system, we often use methods like substitution, elimination, or graphing.

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Most popular questions from this chapter

Solve each system. If a system's equations are dependent or if there is no solution, state this. $$ \begin{aligned} a+b+c &=5 \\ 2 a+3 b-c &=2 \\ 2 a+3 b-2 c &=4 \end{aligned} $$

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The tens digit of a two-digit positive integer is 2 more than three times the units digit. If the digits are interchanged, the new number is 13 less than half the given number. Find the given integer. (Hint. Let \(x=\) the tens-place digit and \(y=\) the units-place digit; then \(10 x+y\) is the number.
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