Questions: Given the Matrices: A=[ [3, -21], [7, 9] ] and B=[ [3, -5], [1, -3] ] Solve: 2A+7B= Multiply [ [7, 4, -3], [4, 5, 0] ][ [7, 4], [-3, 6], [1, -2] ] if possible. If not possible, enter DNE. Graph: y <= x+2, y < -1/4 x -2

Given the Matrices: A=[ [3, -21], [7, 9] ] and B=[ [3, -5], [1, -3] ] Solve: 2A+7B= Multiply [ [7, 4, -3], [4, 5, 0] ][ [7, 4], [-3, 6], [1, -2] ] if possible. If not possible, enter DNE. Graph: y <= x+2, y < -1/4 x -2
Transcript text: Given the Matrices: \[ A=\left[\begin{array}{cc} 3 & -21 \\ 7 & 9 \end{array}\right] \text { and } B=\left[\begin{array}{cc} 3 & -5 \\ 1 & -3 \end{array}\right] \] Solve: \[ 2 A+7 B= \] Multiply $\left[\begin{array}{ccc}7 & 4 & -3 \\ 4 & 5 & 0\end{array}\right]\left[\begin{array}{cc}7 & 4 \\ -3 & 6 \\ 1 & -2\end{array}\right]$ if possible. If not possible, enter DNE. Graph: $\left\{\begin{array}{l}y \leq x+2 \\ y<-\frac{1}{4} x-2\end{array}\right.$
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Solution

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Solution Steps

Step 1: Calculate 2A

Multiply each element of matrix A by 2:

2A = 2 * [[3, -21], [7, 9]] = [[6, -42], [14, 18]]

Step 2: Calculate 7B

Multiply each element of matrix B by 7:

7B = 7 * [[3, -5], [1, -3]] = [[21, -35], [7, -21]]

Step 3: Calculate 2A + 7B

Add the corresponding elements of 2A and 7B:

2A + 7B = [[6, -42], [14, 18]] + [[21, -35], [7, -21]] = [[6+21, -42-35], [14+7, 18-21]] = [[27, -77], [21, -3]]

Final Answer

[[27, -77], [21, -3]]

Step 1: Check if multiplication is possible

The first matrix is a 2x3 matrix and the second is a 3x2 matrix. Since the number of columns in the first matrix (3) equals the number of rows in the second matrix (3), multiplication is possible. The resulting matrix will be a 2x2 matrix.

Step 2: Perform matrix multiplication

[[7, 4, -3], [4, 5, 0]] * [[7, 4], [-3, 6], [1, -2]] = [[(7_7)+(4_-3)+(-3_1), (7_4)+(4_6)+(-3_-2)], [(4_7)+(5_-3)+(0_1), (4_4)+(5_6)+(0_-2)]] = [[49-12-3, 28+24+6], [28-15+0, 16+30+0]] = [[34, 58], [13, 46]]

Final Answer

[[34, 58], [13, 46]]

Step 1: Graph y ≤ x + 2

Graph the line y = x + 2 (slope of 1 and y-intercept of 2). Since it's less than or equal to, the line is solid and the area below the line is shaded.

Step 2: Graph y < -1/4x - 2

Graph the line y = -1/4x - 2 (slope of -1/4 and y-intercept of -2). Since it's strictly less than, the line is dashed and the area below the line is shaded.

Step 3: Find the overlapping shaded region

The solution to the system of inequalities is the overlapping region from the shading of both inequalities.

Final Answer

The graph should show two lines, one solid (y = x + 2) and one dashed (y = -1/4x - 2), with the overlapping shaded region below both lines representing the solution.

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