What type of system is represented by intersecting lines

Active Oldest Votes. This is the "system of equations" point of view. Rugh H. Rugh Sign up or log in Sign up using Google. Sign up using Facebook.

Sign up using Email and Password. Post as a guest Name. Email Required, but never shown. Upcoming Events. Featured on Meta. Learning Objectives Explain what systems of equations can represent. Key Takeaways Key Points A system of linear equations consists of two or more linear equations made up of two or more variables, such that all equations in the system are considered simultaneously.

To find the unique solution to a system of linear equations, we must find a numerical value for each variable in the system that will satisfy all equations in the system at the same time.

Graphically, solutions are points at which the lines intersect. Key Terms system of linear equations : A set of two or more equations made up of two or more variables that are considered simultaneously. Learning Objectives Solve a system of equations in two variables graphically. Key Takeaways Key Points To solve a system of equations graphically, graph the equations and identify the points of intersection as the solutions.

There can be more than one solution to a system of equations. A system of linear equations will have one point of intersection, or one solution. To graph a system of equations that are written in standard form, you must rewrite the equations in slope -intercept form.

Key Terms system of equations : A set of equations with multiple variables which can be solved using a specific set of values. The graphical method : A way of visually finding a set of values that solves a system of equations.

Learning Objectives Solve systems of equations in two variables using substitution. Key Takeaways Key Points A system of equations is a set of equations that can be solved using a particular set of values. The substitution method works by expressing one of the variables in terms of another, then substituting it back into the original equation and simplifying it. It is very important to check your work once you have found a set of values for the variables.

Do this by substituting the values you found back into the original equations. The solution to the system of equations can be written as an ordered pair x , y.

Key Terms substitution method : Method of solving a system of equations by putting the equation in terms of only one variable system of equations : A set of equations with multiple variables which can be solved using a specific set of values. Learning Objectives Solve systems of equations in two variables using elimination. Key Takeaways Key Points The steps of the elimination method are: 1 set the equations up so the variables line up, 2 modify one equation so both equations share a consistent variable that can be eliminated, 3 add the equations together to eliminate the variable, 4 solve, and 5 back-substitute to solve for the other variable.

Always check the answer. This is done by plugging both values into one or both of the original equations. Key Terms elimination method : Process of solving a system of equations by eliminating one variable in order to more simply solve for the remaining variable.

Learning Objectives Explain when systems of equations in two variables are inconsistent or dependent both graphically and algebraically. Key Takeaways Key Points Graphically, the equations in a dependent system represent the same line. The equations in an inconsistent system represent parallel lines that never intersect. We can use methods for solving systems of equations to identify dependent and inconsistent systems: Dependent systems have an infinite number of solutions.

Inconsistent systems have no solutions. Key Terms inconsistent system : A system of linear equations with no common solution because they represent parallel lines, which have no point or line in common.

Learning Objectives Apply systems of equations in two variables to real world examples. Key Takeaways Key Points If you have a problem that includes multiple variables, you can solve it by creating a system of equations. Once variables are defined, determine the relationships between them and write them as equations. Licenses and Attributions. CC licensed content, Shared previously. Notice the results are the same. The general solution to the system is.

The system is dependent so there are infinite solutions of the form. Using what we have learned about systems of equations, we can return to the skateboard manufacturing problem at the beginning of the section. It can be represented by the equation where quantity and price.

The revenue function is shown in orange in Figure. The cost function is the function used to calculate the costs of doing business. It includes fixed costs, such as rent and salaries, and variable costs, such as utilities. The cost function is shown in blue in Figure. The -axis represents quantity in hundreds of units. The y -axis represents either cost or revenue in hundreds of dollars. The point at which the two lines intersect is called the break-even point.

We can see from the graph that if units are produced, the cost is? In other words, the company breaks even if they produce and sell units. They neither make money nor lose money.

The shaded region to the right of the break-even point represents quantities for which the company makes a profit. The shaded region to the left represents quantities for which the company suffers a loss. The profit function is the revenue function minus the cost function, written as Clearly, knowing the quantity for which the cost equals the revenue is of great importance to businesses.

Given the cost function and the revenue function find the break-even point and the profit function. Write the system of equations using to replace function notation. Substitute the expression from the first equation into the second equation and solve for. Then, we substitute into either the cost function or the revenue function. The break-even point is. The profit function is found using the formula. The profit function is. The cost to produce 50, units is? To make a profit, the business must produce and sell more than 50, units.

We see from the graph in Figure that the profit function has a negative value until when the graph crosses the x -axis. Then, the graph emerges into positive y -values and continues on this path as the profit function is a straight line.

This illustrates that the break-even point for businesses occurs when the profit function is 0. The area to the left of the break-even point represents operating at a loss. The cost of a ticket to the circus is for children and for adults. On a certain day, attendance at the circus is and the total gate revenue is How many children and how many adults bought tickets? The total number of people is We can use this to write an equation for the number of people at the circus that day.

The revenue from all children can be found by multiplying by the number of children, The revenue from all adults can be found by multiplying by the number of adults, The total revenue is We can use this to write an equation for the revenue. In the first equation, the coefficient of both variables is 1. We can quickly solve the first equation for either or We will solve for. Substitute the expression in the second equation for and solve for. Substitute into the first equation to solve for. We find that children and adults bought tickets to the circus that day.

Meal tickets at the circus cost for children and for adults. If meal tickets were bought for a total of how many children and how many adults bought meal tickets? Access these online resources for additional instruction and practice with systems of linear equations. Can a system of linear equations have exactly two solutions?

Explain why or why not. If you are solving a break-even analysis and get a negative break-even point, explain what this signifies for the company? This means there is no realistic break-even point.

By the time the company produces one unit they are already making profit. If you are solving a break-even analysis and there is no break-even point, explain what this means for the company.

How should they ensure there is a break-even point? Given a system of equations, explain at least two different methods of solving that system. You can solve by substitution isolating or , graphically, or by addition. For the following exercises, determine whether the given ordered pair is a solution to the system of equations. For the following exercises, graph the system of equations and state whether the system is consistent, inconsistent, or dependent and whether the system has one solution, no solution, or infinite solutions.

For the following exercises, use the intersect function on a graphing device to solve each system. Round all answers to the nearest hundredth. For the following exercises, solve each system in terms of and where are nonzero numbers. Note that and. A stuffed animal business has a total cost of production and a revenue function Find the break-even point.

A fast-food restaurant has a cost of production and a revenue function When does the company start to turn a profit? A cell phone factory has a cost of production and a revenue function What is the break-even point? A musician charges where is the total number of attendees at the concert.

The venue charges? After how many people buy tickets does the venue break even, and what is the value of the total tickets sold at that point? A guitar factory has a cost of production If the company needs to break even after units sold, at what price should they sell each guitar? Round up to the nearest dollar, and write the revenue function. For the following exercises, use a system of linear equations with two variables and two equations to solve.

A number is 9 more than another number. Twice the sum of the two numbers is Find the two numbers. Solutions to systems of equations: dependent vs. Number of solutions to a system of equations. Number of solutions to a system of equations graphically. Practice: Number of solutions to a system of equations graphically.

Number of solutions to a system of equations algebraically.