(Originais Teachy 2023) - Question Easy of Mathematics

In a large engineering system, a building is being designed and energy efficiency is a major concern. To ensure the efficiency of heating and cooling, the air conditioning system must be sized according to the specific needs of the building. The engineer in charge of the project must solve a system of linear equations to determine the ideal operating temperatures. Let the following be the system of equations that models the thermal behavior of the building: 2x + y + 3z = 10, 4x - 6y + 8z = 5, 3x + 7y + 5z = 18. The engineer must rewrite this system of equations in the matrix form Ax = b, where A is the matrix of coefficients, x is the vector of unknowns, and b is the vector of constant terms.

Consider a Cartesian coordinate system where a point P has coordinates (a, b) and a point Q has coordinates (c, d). The absolute value of a number is defined as its distance from the origin of this system. If point P is in the second quadrant and point Q is in the fourth quadrant, with a, b, c, and d being integers, and it is known that the absolute difference between the y coordinates of P and Q is 15, determine the value of a - c, knowing that a < 0 and c > 0.

João wishes to save a certain amount of money every month, and for that, he established a sequence in which the saved amount increases gradually. In the first month, João saved R$ 100.00. In the second month, he saved R$ 120.00, and in the third month, R$ 140.00. Considering that this sequence is arithmetic, (a) determine the reason for this sequence; (b) find the amount that João will save in the fifth month; and (c) calculate the total amount saved by João after the first six months.