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Chapter 8: Problem 85

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

Short Answer

Expert verified
A system of linear equations is a collection of two or more linear equations involving the same set of variables. For example, the equations \(x + y = 5\) and \(2x - 3y = -7\) represent a system of two linear equations in two variables, x and y. The solution to this system, \(x=3, y=2\), is the set of values that satisfy all the equations in the system simultaneously.

Step by step solution

01

Defining a System of Linear Equations

A system of linear equations is a collection of two or more linear equations, each having the same variables. By 'linear', it is implied that all equations in the system are represented graphically as straight lines. The system is composed of equations structured such that the highest power of the variables is one.
02

Explaining a Solution

A solution to a system of linear equations is an ordered pair, (or ordered triple, quadruple, etc., depending on the number of variables), that, when substituted into each equation, makes each equation true.
03

Providing an Example

As an example, consider the system of two linear equations:\[ x + y = 5\]\[2x - 3y = -7\]These two equations form a system of linear equations. A solution to this system is a pair of values (x, y) that makes both equations true when substituted. The solution to this particular system is \(x=3, y=2\)

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Most popular questions from this chapter

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 $$\begin{aligned} &z=3 x+2 y\\\ &\left\\{\begin{array}{c} {x \geq 0, y \geq 0} \\ {2 x+y \leq 8} \\ {x+y \geq 4} \end{array}\right. \end{aligned}$$

An object moves in simple harmonic motion described by \(d=6 \cos \frac{3 \pi}{2} t,\) where \(t\) is measured in seconds and \(d\) in inches. Find: a. the maximum displacement b. the frequency c. the time required for one cycle. (Section \(5.8, \text { Example } 8)\)

Determine whether each statement makes sense or does not make sense, and explain your reasoning. When graphing \(3 x-4 y<12\), it's not necessary for me to graph the linear equation \(3 x-4 y=12\) because the inequality contains a \(<\) symbol, in which equality is not included.

Sketch the graph of the solution set for the following system Of inequalities: $$ \left\\{\begin{array}{l} {y \geq n x+b(n<0, b>0)} \\ {y \leq m x+b(m>0, b>0)} \end{array}\right. $$

Will help you prepare for the material covered in the next section. In each exercise, graph the linear function. $$ f(x)=-2 $$
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