M_2x3 x N_3x4 = P_2x4 

In general matrix notation, M_rxc shows that the matrix is named M and r is the number of rows and c is the number of columns.  When multiplying two matrices, the number of columns in the first matrix must match the number of rows in the second matrix.  In addition, when adding or subtracting matrices, the matrices must be of the same size.

In order to solve the matrix, the determinant rule "ad-bc" must be used. \(\displaystyle 8\) is in the "a" position, \(\displaystyle 3\) is in the "b" position, \(\displaystyle -2\) is in the "c" position, and \(\displaystyle 4\) is in the "d" position. After plugging the numbers into "ad-bc," we get

\(\displaystyle (8\times 4)-(-2\times 3)=(32)-(-6)=32+6=38\)

To set up and augmented matrix for a 3x3 system of equations, all equations must be in standard form \(\displaystyle Ax+By + Cz= D\). The third equation is already in standard form; the first two are not and must be rewritten as such.

\(\displaystyle y = 5x\)

\(\displaystyle y - y = 5x - y\)

\(\displaystyle 5x - y = 0\)

\(\displaystyle x+ y = 3z\)

\(\displaystyle x+ y- 3z = 3z - 3z\)

\(\displaystyle x+ y- 3z = 0\)

The system is now

\(\displaystyle 5x - y = 0\)

\(\displaystyle x+ y- 3z = 0\)

\(\displaystyle 2x - 3y + 4z = 100\)

Write the augmented matrix with each row comprising the coefficients of one equation in order:

\(\displaystyle \begin{bmatrix} 5 & -1 & 0 \\ 1 & 1 & -3 \\2 & - 3 & 4 \end{matrix} \begin{vmatrix} 0 \\0\\ 100 \end{bmatrix}\)

is the correct choice.

To set up and augmented matrix for a 2x2 system of equations, both equations must be in standard form \(\displaystyle Ax+By = C\). The second equation is already in standard form.

Rewrite the first equation in standard form as follows:

\(\displaystyle y = 4x + 7\)

\(\displaystyle 4x + 7 = y\)

\(\displaystyle 4x + 7- y - 7 = y - y - 7\)

\(\displaystyle 4x - y = - 7\)

The system has been rewritten as

\(\displaystyle 4x - y = - 7\)

\(\displaystyle 2x - 7y = 8\)

Write the augmented matrix with each row comprising the coefficients of one equation in order:

\(\displaystyle \begin{bmatrix} 4&-1 \\ 2& -7 \end{matrix} \begin{vmatrix} -7\\ 8 \end{bmatrix}\)

is the correct choice.

If we let \(\displaystyle x\), \(\displaystyle y\), and \(\displaystyle z\) represent the number of large, medium, and small boxes sold, respectively, since twenty more medium boxes than large were sold, one equation of the 3x3 system will be

\(\displaystyle y = x+20\)

or, in standard form,

\(\displaystyle x-y = -20\)

Since 445 boxes were sold, another linear equation will be

\(\displaystyle x+y+z = 445\)

The money raised from the sale of \(\displaystyle x\) large boxes of cookies, each of which cost $5.75, is \(\displaystyle 5.75x\); the money raised from the sale of \(\displaystyle y\) small boxes of cookies, each of which cost $4.75, is \(\displaystyle 4.75y\); and the money raised from the sale of \(\displaystyle z\) small boxes of cookies, each of which cost $3.25, is \(\displaystyle 3.25z\). 

The total money raised is $1,924.25, so the other linear equation of the system is 

\(\displaystyle 5.75x+ 4.75y+ 3.25z= 1,924.25\)

The augmented matrix of this system will comprise the coefficients of these equations, all of which are now standard form, so the matrix will be

\(\displaystyle \begin{bmatrix} 1 & -1 & 0 \\ 1 & 1 & 1 \\5.75 & 4.75 & 3.25 \end{matrix} \begin{vmatrix} -20 \\445\\ 1,924.25\end{bmatrix}\),

which is the correct choice.

If we let \(\displaystyle x\) and \(\displaystyle y\) represent the number of large and small boxes sold, respectively, since 305 boxes were sold, one linear equation of the 2x2 system will be

\(\displaystyle x+y = 305\)

The money raised from the sale of \(\displaystyle x\) large boxes of cookies, each of which cost $4.75, is \(\displaystyle 4.75x\); the money raised from the sale of \(\displaystyle y\) small boxes of cookies, each of which cost $3.25, is \(\displaystyle 3.25y\). The total money raised is $1,196.75, so the other linear equation of the system is 

\(\displaystyle 4.75x+ 3.25y=1,196.75\)

The augmented matrix of this system will comprise the coefficients of these equations, both of which are in standard form, so the matrix will be

\(\displaystyle \begin{bmatrix} 1&1 \\ 4.75&3.25 \end{matrix} \begin{vmatrix} 305 \\ 1,196.75 \end{bmatrix}\)

If we let \(\displaystyle x\) and \(\displaystyle y\) represent the amount of the weaker and stronger solutions, respectively, since the chemist needs 1 liter of the resulting solution, one linear equation of the 2x2 system will be 

\(\displaystyle x+y = 1\)

The amount of alcohol in \(\displaystyle x\) liters of a 10% solution will be \(\displaystyle 0.1x\); the amount of alcohol in \(\displaystyle y\) liters of a 40% solution will be \(\displaystyle 0.4y\); and the total amount of alcohol in the resulting solution will be 20 % of a liter, or 0.20 liters. Therefore, the second linear equation of the system will be

\(\displaystyle 0.1x + 0.4y = 0.2\)

The augmented matrix of this system will comprise the coefficients of these equations, both of which are in standard form, so the matrix will be

\(\displaystyle \begin{bmatrix} 1&1 \\ 0.1 &0.4 \end{matrix} \begin{vmatrix} 1 \\0.2 \end{bmatrix}\),

which is the correct choice.

To find the number of blue items Jon has, we sum the entries in the column labelled \(\displaystyle \text{blue}\) and get \(\displaystyle 2+5=7\). Recall that matrices are organized by rows and columns where each entry refers to the number of items that are, in this case, both of the same color AND type. All the entries in any one column are of the same color. All of the entries in any row are of the same clothing type.  

If Jon has three red shirts and two blue pants. The total number of combinations of one shirt and one pair of pants he can wear is going to be the number of shirts he can wear with the first pair of pants, \(\displaystyle 3\), plus the number of shirt he can wear with the second pair of pants, also \(\displaystyle 3\). That sums to \(\displaystyle 6\) total outfits.

If the barista mixes \(\displaystyle x\) pounds of Peppermint Nirvana and \(\displaystyle y\) pounds of Strawberry Fields to make twenty pounds of tea total, then

\(\displaystyle x+y = 20\)

will be one of the equations in the system.

\(\displaystyle x\) pounds of Peppermint Nirvana tea for $12 a pound will cost a total of \(\displaystyle 12x\) dollars; \(\displaystyle y\) pounds of Strawberry Fields tea will cost a total of \(\displaystyle 15y\) dollars. Tewnty pounds of the Strawberry Peppermint Delight tea for $13 a pound will cost \(\displaystyle 13 \times 20 = 260\) dollars. Since the tea will sell for the same price blended as separate, the other equation of the system will be

\(\displaystyle 12x + 15y = 260\)

The augmented matrix of this system will comprise the coefficients of these equations, both of which are in standard form, so the matrix will be

\(\displaystyle \begin{bmatrix} 1&1 \\ 12 & 15 \end{matrix} \begin{vmatrix} 20 \\ 260 \end{bmatrix}\).