Question

This system of equations has been placed in a matrix:

y = 650x + 175
y = 25,080 − 120x
Column 1 Column 2 Column 3


Row 1 _ -1 _


Row 2 75 _ _

Answer 1

Column 1: 75

Column 2: -1

Column 3: -650

Step-by-step explanation:

The given system of equations can be represented as a matrix using the coefficients of the variables x and y. The coefficients for x and y in the first equation y = 650x + 175 are 650 and -1, respectively, forming the first row of the matrix. Similarly, for the second equation y = 25,080 − 120x, the coefficients become -120 and 1, forming the second row of the matrix.

In the matrix representation, the coefficients of x are placed in Column 1, those of y in Column 2, and the constant terms in Column 3. Therefore, for the given system, Column 1 corresponds to the coefficients of x, where 75 and -120 are the coefficients for x in the first and second equations, respectively. Column 2 contains the coefficients of y, which are -1 and 1 for the first and second equations, respectively. Column 3 holds the constants, which are -650 and 25,080 for the first and second equations, respectively.

Understanding how to organize coefficients in a matrix helps in solving systems of equations using matrix methods, such as Gaussian elimination or matrix inversion, making calculations more efficient and systematic.

Answer 2

The system of equations can be represented by the matrix equation A*x = b, where A is the coefficient matrix, x is the array of unknowns, and b is the array of right-hand sides of the equations. In this case, the coefficient matrix is:

A = [[650, -1, 0],

[75, 0, 1],

[0, -120, 1]]

And the vector of right-hand sides is:

b = [650, 25080, -120]

To solve this system of equations, we need to find the value of x that satisfies the equation A*x = b. This can either be done by manual methods or by using software such as MATLAB.

Manual methods include solving for x using row-reduction techniques, such as Gaussian elimination, or using an algebraic formula for solving linear systems.

Using MATLAB, we can enter the coefficient matrix A and the vector b, create the system of equations, and solve for x using a built-in function:

A = [[650, -1, 0];

[75, 0, 1];

[0, -120, 1]];

b = [650;

25080;

-120];

x = linsolve(A,b);

format long;

disp(x);

This will output x = [0, 25.2142857143, 1.5714285714].

Alternatively, we can use an online calculator such as Wolfram|Alpha to solve the system of equations:

less

A = {{650, -1, 0}, {75, 0, 1}, {0, -120, 1}}

b = {650, 25080, -120}

Solve[A . {x, y, z} == b, {x, y, z}]

ans = {x -> 0, y -> 25.2143, z -> 1.57143}

This will output x = 0,