For the following exercises, set up the augmented matrix that describes the situation, and solve

For the following exercises, set up the augmented matrix...

Key Concepts

Augmented Matrix

An augmented matrix is a compact way of representing a system of linear equations, where the coefficients of the variables and the constants from the right-hand side of the equations are arranged in a matrix. This format simplifies operations like row reduction, which are used to solve the system systematically.

System of Linear Equations

A system of linear equations consists of two or more equations that share similar variables. In contexts like break-even analysis, the system is used to equate different linear relationships, such as revenue and cost, to find the point where they are equal.

Break-even Analysis

Break-even analysis is the study of the point at which revenue generated equals the costs incurred, resulting in no net loss or gain. It involves setting up and solving equations that represent income and expenditure in order to determine the necessary quantity of items produced or sold to cover costs.

Transcript

00:01 Okay, so this one has us using a system of equations to discover the break -even point, and it involves selling bicycles and producing of bicycles.

00:12 So the first thing we should do is start off with what we know.

00:16 And we know, based on the problem, that we can sell a bike for $250.

00:25 The cost to produce the bike is $180 for each bike.

00:32 And there is an initial startup cost in the amount of $3 ,500.

00:44 So when we generate our system of equations, we're going to come up with two equations, and one of those equations is going to represent the costs associated with producing of the bike.

00:55 And the other equation is going to be the money generated or the revenues generated from the sale of the bike.

01:03 So for our cost, i'm going to use c of x to represent cost.

01:10 We're using our function notation.

01:13 And there is an initial $3 ,500 and then an additional $180 for each bike we produce.

01:22 So since we used the variable x, we should define what x is.

01:26 X represents the number of bikes.

01:38 And our second equation is generated by the sale of the bike.

01:42 So we're going to call it r of x for the revenue coming in, and we end up with $250 for every bike we sell.

01:52 Now that we have our system of equations, we have to discuss the concept of break -even, and our break -even point is where the costs associated with production and the costs that are generated from the sale end up zeroing each other out.

02:11 So in other words, we've got the cost of production, equaling the revenues that we brought in.

02:22 Now this problem also asked us to use augmented matrices and gaussian elimination to solve.

02:28 And before we can transition into the augmented matrix, we're going to have to replace the c of x with the letter or the variable y.

02:38 Both of them represent the output.

02:41 So we are going to rewrite our system as y equals 3 ,500 plus 180x.

03:05 And because c of x and r of x are equal at the break -even point, in place of r of x we can use the same variable y.

03:21 Now before we can use augmented matrices, we also need to rewrite that system of equations in standard form.

03:30 So our standard form is when we get the, the x and the y variables together on one side of the equal sign and the constants on the other.

03:41 So the first equation can be rewritten as 180x minus y equals negative 3 ,500.

03:54 And the second equation 250x minus y equals zero.

04:02 So we are ready now to transition into our augmented matrix.

04:07 And because there are two equations, our matrix is going to have two rows.

04:14 And on the left side of the matrix, we would use the coefficients.

04:19 And our coefficients are 180, and there is a negative 1 in front of that y.

04:25 So we would have 180 and negative 1.

04:30 And on the right side of the matrix, we would put our answer.

04:34 And the answer to the first equation is negative 3 ,000.

04:41 500.

04:43 So the second equation, the coefficients are 250 and negative 1, and the answer was 0.

04:55 So we have now set up the augmented matrix associated with our system of equations.

05:02 When it comes time to use gaussian elimination in order to solve the augmented matrix, our ultimate goal is to get the matrix to look like this.

05:16 A one zero and a zero one on the left and numbers on the right.

05:24 So we're trying to get our system of equations into that format...

Please give Ace some feedback