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

For the following exercises, set up the augmented matrix that describes the situation, and solve for the desired solution. A bag of mixed nuts contains cashews, pistachios, and almonds. Originally there were 900 nuts in the bag. 30% of the almonds, 20% of the cashews, and 10% of the pistachios were eaten, and now there are 770 nuts left in the bag. Originally, there were 100 more cashews than almonds. Figure out how many of each type of nut was in the bag to begin with.

For the following exercises, set up the augmented matrix that describes the situation, and solve for the desired solution.
A bag of mixed nuts contains cashews, pistachios, and almonds. Originally there were 900 nuts in the bag. 30% of the almonds, 20% of the cashews, and 10% of the pistachios were eaten, and now there are 770 nuts left in the bag. Originally, there were 100 more cashews than almonds. Figure out how many of each type of nut was in the bag to begin with.

Key Concepts

Translating_Word_Problems_into_Equations

This key concept focuses on interpreting verbal descriptions and translating them into mathematical equations. It involves identifying the variables, understanding the relationships and constraints described, and then formulating equations that accurately represent these conditions.

Gaussian_Elimination

This is a method used to solve systems of linear equations by systematically performing row operations on the augmented matrix. The goal is to reduce the matrix to an upper triangular form (or further into reduced row echelon form), making it easier to back-substitute and find the values of the unknown variables.

Systems_of_Linear_Equations

This concept involves setting up multiple equations that represent different relationships or constraints in a problem. In many word problems, individual statements are converted into linear equations that collectively describe how unknown quantities relate to each other, forming a system of equations that must be solved simultaneously.

Augmented_Matrix

An augmented matrix is a compact way to represent a system of linear equations, where the coefficients of the variables and the constants are arranged in a structured manner. It is particularly useful for applying systematic methods, such as Gaussian elimination, to simplify and solve the problem efficiently.

Transcript

00:01 Okay, so in the following problem, we have a bag full of mixed nuts, which contains almonds, cashews, and pistachios.

00:08 So at the beginning, there were 900 nuts.

00:12 And after we have eaten 30 % of the almonds, 20 % of the cashews, and 10 % of the pistachios, we have a total of 770 nuts.

00:23 Now, at the beginning, there were 100 more cashews than almonds.

00:27 And with this information, we want to find the original amount of each type of.

00:31 Of knot that was on the back.

00:35 So we can start off by saying that x is going to be equal to cashews, y is going to be equal to pistachios, and then z is going to be equal to all months.

00:54 We know that x plus y plus z at the beginning was 900.

00:59 So x plus y plus z is equal to 900 because now this is the beginning and we have 100 % of our casios, 100 % of our pistachios and 100 % of our of our almonds.

01:14 Now when we have eaten 30 % of the almonds, we have a grand total, we have 70 % left.

01:22 So we have 0 .7 z.

01:26 And now we know we also, so we, i want to rewrite this.

01:30 So we have 0 .7 z.

01:33 After we have eaten 20 % of our cashios, we have 80 % left.

01:38 So 0 .8.

01:40 And cashios is represented by x and after we have eaten 10 % of our pistachios we have left a 0 .9 % 90 % left which is also 0 .9 z sorry why and all of these quantities must equal 770 now we also know that at the beginning casios has a hundred more knots than almonds so in this case we can say that x is equal to z plus a hundred and we can rewrite this as x minus z is equal to a hundred.

02:28 So with these three equations, you can actually start off by writing an augmented matrix.

02:36 So this matrix will look like one, one, one, 900, zero point seven, oh, sorry, 0 .8, 0 .9, and 0 .7, and 7, and 770.

02:55 And finally, 1, 0, negative 1, and 100.

03:00 Now we can perform row operations with this matrix right here to determine the amount of knots that were of each type of nut that was in the beginning.

03:10 So the first row operation that we're going to perform is to multiply row 1 by 0 .8 and we're going to subtract that from row 2.

03:22 Sorry, we're going to multiply this by a negative 0 .8 and added to row 2...

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