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Chapter 5: Problem 91

When using the addition or substitution method, how can you tell if a system of linear equations has no solution? What is the relationship between the graphs of the two equations?

Short Answer

Expert verified
For a system of linear equations, if the addition or substitution method leads to a false statement, then the system has no solution. Graphically, such a system is represented by two parallel lines which do not intersect.

Step by step solution

01

Identify The Statement Resulting from the Methods

In both the addition and substitution method for solving a system of linear equations, you need to manipulate the equations until you obtain a new equation or a definitive statement. If this process leads to a false statement such as 0=1, it suggests that the system of equations has no solutions.
02

Understand Graphical Interpretation

When a system of linear equations has no solution, their respective graphs are parallel lines. Two parallel lines never intersect hence they do not have a common solution.
03

Linkage Between Step 1 and Step 2

To tie it all together, when handling a system of linear equations, a false statement resulting from substitution or addition method indicates a system with no solution. Graphically, such a system is represented by parallel lines that do not intersect.

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Most popular questions from this chapter

Describe a number of business ventures. For each exercise, a. Write the cost function, \(C\). b. Write the revenue function, \(R\). c. Determine the break-even point. Describe what this means. You invested \(\$ 30,000\) and started a business writing greeting cards. Supplies cost \(2 \notin\) per card and you are selling each card for \(50 \mathrm{e}\). (In solving this exercise, let \(x\) represent the number of cards produced and sold.)

What is a system of linear equations? Provide an example with your description.

Exercises 37-39 will help you prepare for the material covered in the first section of the next chapter. Consider the following array of numbers: $$\left[\begin{array}{rrr}1 & 2 & -1 \\ 4 & -3 & -15\end{array}\right]$$ Rewrite the array as follows: Multiply each number in the top row by \(-4\) and add this product to the corresponding number in the bottom row. Do not change the numbers in the top row.

What is a solution of a system of linear inequalities?

In Exercises \(5-14,\) an objective function and a system of linear inequalities representing constraints are given. a. Graph the system of inequalities representing the constraints. b. Find the value of the objective function at each corner of the graphed region. c. Use the values in part ( \(b\) ) to determine the maximum value of the objective function and the values of \(x\) and \(y\) for which the maximum occurs. Objective Function Constraints $$\begin{aligned}&z-4 x+2 y\\\&\left\\{\begin{array}{l}x \geq 0, y \geq 0 \\\2 x+3 y \leq 12\end{array}\right.\\\&\begin{array}{l}3 x+2 y \leq 12 \\\x+y \geq 2\end{array}\end{aligned}$$
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