Question

((HELP ASAP RN PLEASE))
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.

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

Answer:

Step-by-step explanation:

To solve the given problem using matrices, let's represent the number of ounces of jam in each type of jar as variables.

Let:

L = Number of ounces of jam in the large jar

S = Number of ounces of jam in each small jar

Based on the given information, we can create the following equations:

Equation 1: L + 3S = 14 (One large jar and three small jars together can hold 14 ounces of jam)

Equation 2: L - S = 2 (One large jar minus one small jar can hold 2 ounces of jam)

We can represent these equations in matrix form as follows:

| 1   3 |   | L |   | 14 |

| 1  -1 | * | S | = |  2 |

To solve the system of equations, we can use matrix inversion. First, let's calculate the inverse of the coefficient matrix:

| 1   3 |   | a   -3 |

| 1  -1 | = | 1   -1 |

Next, multiply the inverse of the coefficient matrix by the constant matrix:

| a   -3 |   | 14 |   | L |

| 1   -1 | * |  2 | = | S |

Performing the matrix multiplication, we get:

(14a - 3*2) = L

(a - 2) = S

Simplifying the equations, we have:

14a - 6 = L

a - 2 = S

Now, we can substitute the value of "a" into the equation for "L" to solve for "L":

14a - 6 = L

14(a - 2) - 6 = L

14a - 28 - 6 = L

14a - 34 = L

So, we have found that L = 14a - 34.

Now, we can substitute the value of "a" into the equation for "S" to solve for "S":

a - 2 = S

Since we don't have a specific value for "a," we can express "S" in terms of "a" as well:

S = a - 2

So, we have found that S = a - 2.

Therefore, the solution for the system of equations is L = 14a - 34 and S = a - 2, where "a" represents any real number.

This means that the number of ounces of jam in the large jar can be represented as 14 times any real number "a" minus 34, and the number of ounces of jam in the small jars can be represented as any real number "a" minus 2. The specific values for L and S would depend on the chosen value of "a".

Answer 2

Answer:

• large: 5 oz
• small: 3 oz

Step-by-step explanation:

You want a matrix solution to the equations represented by ...

  $\left[\begin{array}{cc}1&3\\1&-1\end{array}\right] \left[\begin{array}{c}l\\s\end{array}\right] =\left[\begin{array}{c}14\\2\end{array}\right]$

Augmented matrix

Adding the coefficients as a column in the coefficient matrix, we get the augmented matrix ...

  $\left[\begin{array}{cc|c}1&3&14\\1&-1&2\end{array}\right]$

Row reduction

Replacing the second row with the difference of the first and second rows, we have ...

  $\left[\begin{array}{cc|c}1&3&14\\0&4&12\end{array}\right]$

This upper triangular form is sufficient to tell the solution. The remaining steps formalize that solution.

Dividing the second row by 4 gives ...

  $\left[\begin{array}{cc|c}1&3&14\\0&1&3\end{array}\right]$

Finally, subtracting 3 times the second row from the first gives the solution.

  $\left[\begin{array}{cc|c}1&0&5\\0&1&3\end{array}\right]$

This tells us

  $\left[\begin{array}{c}l\\s\end{array}\right] =\left[\begin{array}{c}5\\3\end{array}\right]$

A large jar holds 5 ounces of jam; a small jar holds 3 ounces.

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Additional comment

We could also solve this by computing the inverse of the coefficient matrix, and left-multiplying each side of the equation by that.

We like the calculator's ability to write the augmented matrix in "reduced row-echelon form" in one step. Using matrix multiplication requires more work, even with the calculator's help.

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