Solve the problem.A mathematician has found 5 solutions to a homogeneous system of 40 equations in 42 variables. The 5 solutions are linearly independent and all other solutions can be constructed by adding together appropriate multiples of these 5 solutions. Will the system necessarily have a solution for every possible choice of constants on the right side of the equation? Explain.
What will be an ideal response?
No.
Let A be the 40 × 42 coefficient matrix of the system. The 5 solutions are linearly independent and span Nul A, so dim Nul A = 5. By the Rank Theorem, dim Col A = 42 - 5 = 37. Since 37 < 40, Col A does not span ?40. So not every nonhomogeneous equation Ax = b has a solution.
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The stack plot below shows the value of each of Danny's investments. The stack plot contains three regions. The uppermost unshaded region represents the value of Danny's investment in individual stocks. The center shaded region represents the value of Danny's investment in mutual funds and the bottom region in black represents the value of Danny's investment in a CD. The thickness of a region at a particular time tells you its value at that time. 
Use the graph to answer the question.In which year was the value of Danny's investment in individual stocks the least?
A. year 0 B. year 7 C. year 1 D. year 8
Provide an appropriate response.When two linear equations are solved for y, the slopes are equal and the y-intercepts are different, the lines are _____.
A. parallel lines B. intersecting lines C. the same line
Use the accompanying graph of y = f(x) to sketch the graph of the indicated equation.y = - 3f(x + 4) + 3
A. 
B. 
C. 
D. 
Factor the greatest common factor from the polynomial. Assume any variable exponents represent whole numbers.15xy + 21xz - 6xw
A. 3(5xy + 7xz - 2xw) B. 3x(5y + 7z - 2w) C. x(15y + 21z - 6w) D. x(15xy + 21xz - 6xw)