5) How many solutions does the system of equations have?x-4y=12 and 5x-20y=60

A.one
B.two
C.infinitely many
D.none
6) How many solutions does the system of equations have? y-6x=-3 and 4y-24x=-16
A.one
B.two
C.infinitely many
D.none

I need help can someone help me.

To determine the number of solutions for each system of equations, we can use the method of solving them. This method involves creating an augmented matrix and performing row operations to find the row echelon form or the reduced row echelon form of the matrix.

For the first system of equations:

x - 4y = 12
5x - 20y = 60

To solve this system, let's write it in the form of an augmented matrix:

[1 -4 | 12]
[5 -20 | 60]

By performing row operations, we can simplify the augmented matrix to row echelon form:

[1 -4 | 12]
[0 0 | 0]

The row echelon form tells us that there is a dependent variable (y) and an independent variable (x) that can be expressed in terms of the other. Since there are infinitely many valid values of x and y that satisfy this system, the answer is C. Infinitely many solutions.

For the second system of equations:

y - 6x = -3
4y - 24x = -16

Again, let's write it in the form of an augmented matrix:

[-6 1 | -3]
[-24 4 | -16]

By performing row operations, we can simplify the augmented matrix to row echelon form:

[1 -1/6 | 1/2]
[0 0 | 0]

Similar to the first system, the row echelon form reveals that there is a dependent variable (y) and an independent variable (x). Since there are infinitely many valid values of x and y that satisfy this system, the answer is also C. Infinitely many solutions.

what is the answer for 5 and 6

A.one
B.two
C.infinitely many
D. none

so whats the anser to 5

thank you so much. thanks

5) Multiply first equation by 5.

5x - 20y = 60

6) Multiply first equation by 4.

4y - 24x = -12

Subtracting first equation from the second, what does that tell you?

come on. For #5, you have

5x-20y=60
5x-20y=60
They are the same equation. So, any solution to one is also a solution to the other.

Try a similar comparison for #6 and just think about what it means.