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Chapter 5: 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 linear equations involving the same set of variables. An example is \[\begin{{align*}}3x + 2y &= 12, \ x - y &= 1.\end{{align*}}\]

Step by step solution

01

Definition of A System of Linear Equations

A system of linear equations (or linear system) is a collection of two or more linear equations involving the same set of variables. For example, with two variables x and y, a linear system is:
02

Giving an Example

An example of a system of linear equations is: \[\begin{{align*}}3x + 2y &= 12, \x - y &= 1.\end{{align*}}\]where x and y are the variables. To satisfy both equations, the values for x and y need to satisfy both conditions, as given by these two equations.
03

Explanation of the Example

In the given example, the system consists of two equations and two variables (x and y). It suggests two conditions for the variables. The first equation states that if 3 units of x are combined with 2 units of y, we get 12 units. The second equation states that if you take one unit of x and subtract one unit of y from it, you get 1 unit. Both conditions must hold true simultaneously.

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Most popular questions from this chapter

What is a system of linear inequalities?

A television manufacturer makes rear-projection and plasma televisions. The profit per unit is \(\$ 125\) for the rear-projection televisions and \(\$ 200\) for the plasma televisions. a. Let \(x=\) the number of rear-projection televisions manufactured in a month and let \(y=\) the number of plasma televisions manufactured in a month. Write the objective function that models the total monthly profit. b. The manufacturer is bound by the following constraints: \(\cdot\) Equipment in the factory allows for making at most 450 rear-projection televisions in one month. \(\cdot\) Equipment in the factory allows for making at most 200 plasma televisions in one month. \(\cdot\) The cost to the manufacturer per unit is \(\$ 600\) for the rear- projection televisions and \(\$ 900\) for the plasma televisions. Total monthly costs cannot exceed \(\$ 360,000\). Write a system of three inequalities that models these constraints. c. Graph the system of inequalities in part (b). Use only the first quadrant and its boundary, because \(x\) and \(y\) must both be nonnegative. d. Evaluate the objective function for total monthly profit at each of the five vertices of the graphed region. [The vertices should occur at \((0,0),(0,200),(300,200),(450,100),\) and \((450,0) .]\) e. Complete the missing portions of this statement: The television manufacturer will make the greatest profit by manufacturing- rear-projection televisions each month and maximum monthly profit is $\$$

Explain how to solve a system of equations using the substitution method. Use \(y-3=3 x\) and \(3 x+4 y=6\) to illustrate your explanation.

Use a system of linear equations to solve. Looking for Mr. Goodbar? It's probably not a good idea if you want to look like Mr. Universe or Julia Roberts. The graph shows the four candy bars with the highest fat content, representing grams of fat and calories in each bar. Basedon the graph. (GRAPH CAN'T COPY) A collection of Halloween candy contains a total of 12 Snickers bars and Reese's Peanut Butter Cups. Chew on this: The grams of fat in these candy bars exceed twice the daily maximum desirable fat intake of 70 grams by 26.5 grams. How many bars of each kind of candy are contained in the Halloween collection?

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I use the same steps to solve nonlinear systems as I did to solve linear systems, although I don't obtain linear equations when a variable is eliminated.
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