Question

URGENT!!! pls help :)

Question 1 (Essay Worth 4 points)

One large jar and three small jars together can hold 14 ounces of jam. One large jar minus one small jar can hold 2 ounces of jam.

A matrix with 2 rows and 2 columns, where row 1 is 1 and 3 and row 2 is 1 and negative 1, is multiplied by matrix with 2 rows and 1 column, where row 1 is l and row 2 is s, equals a matrix with 2 rows and 1 column, where row 1 is 14 and row 2 is 2.

Use matrices to solve the equation and determine how many ounces of jam are in each type of jar. Show or explain all necessary steps.

Answer 1

To solve this problem, we can use matrices. Let's call the number of ounces of jam in the large jar "L" and the number of ounces of jam in each small jar "S". We can set up two equations based on the information given:

L + 3S = 14 (since one large jar and three small jars together can hold 14 ounces of jam)

L - S = 2 (since one large jar minus one small pot can hold 2 ounces of jam)

We can write these equations in matrix form:

[1 3] [L] [14]

[1 -1] [S] = [2]

To solve for L and S, we need to multiply both sides of the equation by the inverse of the matrix on the left:

[1 3]^-1 [1 3] [L] [1 3]^-1 [14]

[1 -1] [1 -1] [S] = [1 -1] [2]

The inverse of the matrix [1 3; 1 -1] is:

[1/4 3/4]

[1/4 -1/4]

So we have:

[L] [1/4 3/4] [14]

[S] = [1/4 -1/4] [2]

Multiplying out the matrices gives us the following:

[L] [(1/4)*14 + (3/4)*2]

[S] = [(1/4)*14 - (1/4)*2]

So L = 5 and S = 3. Therefore, there are 5 ounces of jam in the large jar and 3 ounces in each small jar.

Answer 2

the large jar contains 5 ounces of jam, and each small jar contains 3 ounces of jam.

Explanation:

To solve the given problem using matrices, let's assign variables to represent the number of ounces of jam in each type of jar. We'll use the following variables:

L: Ounces of jam in the large jar.

S: Ounces of jam in each small jar.

Now, let's set up the equations based on the information given:

Equation 1: One large jar and three small jars together can hold 14 ounces of jam.

This equation can be written as:

1L + 3S = 14

Equation 2: One large jar minus one small jar can hold 2 ounces of jam.

This equation can be written as:

1L - 1S = 2

Now, let's represent these equations in matrix form:

Equation 1:

[1 3] [L] [14]

*

Equation 2:

[1 -1] [S] [ 2]

Multiplying the matrices gives us:

[1L + 3S] [14]

=

[1L - 1S] [ 2]

Simplifying the matrix equation, we have:

[1L + 3S] = [14]

[1L - 1S] = [ 2]

This can be written as a system of equations:

1L + 3S = 14 --(Equation A)

1L - 1S = 2 --(Equation B)

To solve this system, we can use the method of elimination. Let's eliminate the variable L by adding Equation A and Equation B:

(Equation A) + (Equation B):

1L + 3S + 1L - 1S = 14 + 2

Simplifying:

2L + 2S = 16 --(Equation C)

Now, we have two equations:

2L + 2S = 16 --(Equation C)

1L - 1S = 2 --(Equation B)

Let's multiply Equation B by 2 to make the coefficients of L in both equations equal:

2(1L - 1S) = 2 * 2

2L - 2S = 4 --(Equation D)

Now, we have two equations:

2L + 2S = 16 --(Equation C)

2L - 2S = 4 --(Equation D)

We can now eliminate the variable S by adding Equation C and Equation D:

(Equation C) + (Equation D):

2L + 2S + 2L - 2S = 16 + 4

Simplifying:

4L = 20

Dividing both sides of the equation by 4, we get:

L = 5

Now, substitute the value of L back into Equation B to find S:

1L - 1S = 2

1(5) - 1S = 2

5 - 1S = 2

-1S = 2 - 5

-1S = -3

Dividing both sides of the equation by -1, we get:

S = 3

Therefore, the large jar contains 5 ounces of jam, and each small jar contains 3 ounces of jam.