Question

3. How many solutions does the system of equations have?

4x = -12y + 16 and x + 3y = 4

a.) one
b.) two
c.) infinitely many
d.) none

4.) How many solutions does the system of equations have?

y = 6x + 2 and y = 6x + 4

a.) one
b.) two
c.) infinite
d.) none

5.) How many solutions does the system of equations have?

x - 2y = 6 and 3x - 6y = 18

a.) one
b.) two
c.) infinitely many
d.) none.

6.) How many solutions does the system of equations have?
y - 7x = -14 and 7y - 49x = -2
a.) one
b.) two
c.) infinitely many
d.) none

I don't know how to do these, and explanation of how these are done would be nice.

Answer 1

Part 3) Option C infinitely solutions

Part 4) Option D none

Part 5) Option C infinitely solutions

Part 6) Option D none

Step-by-step explanation:

To determine the number of solutions for a system of equations, we need to solve the equations simultaneously and see how many common solutions they have.

Let's go through each system of equations:

System 1:

4x = -12y + 16 and x + 3y = 4

By rearranging the second equation, we can rewrite it as x = 4 - 3y.

Substituting x in the first equation, we get 4(4 - 3y) = -12y + 16.

Simplifying, we have 16 - 12y = -12y + 16.

This equation simplifies to 0 = 0, which means the two equations are equivalent and represent the same line. Therefore, the system has infinitely many solutions (c).

System 2:

y = 6x + 2 and y = 6x + 4

Since both equations have the same slope, but different y-intercepts, the lines represented by the equations are parallel. Parallel lines never intersect, so the system has no solution (d).

System 3:

x - 2y = 6 and 3x - 6y = 18

Dividing the second equation by 3, we get x - 2y = 6.

This shows that the two equations represent the same line. Therefore, the system has infinitely many solutions (c).

System 4:

y - 7x = -14 and 7y - 49x = -2

Multiplying the first equation by -7, we get -7y + 49x = 98.

Equation A and equation B are parallel lines, because has the same slope.

The slope is equal to m=7 therefore the system has no solution.

Answer 2

Part 3) Option C infinitely solutions

Part 4) Option D none

Part 5) Option C infinitely solutions

Part 6) Option D none

Step-by-step explanation:

Part 3) we have

`4x=-12y+16` -----> convert to standard form

`4x+12y=16` -----> equation A

`x+3y=4` -----> equation B

Multiply the equation B by
`4`

`4(x+3y)=4*4` ------>
`4x+12y=16`

The equation A and the equation B are the same

therefore

The system has infinitely solutions

Part 4) we have

`y=6x+2` ------> equation A

`y=6x+4` ------> equation B

equation A and equation B are parallel lines, because has the same slope

the slope is equal to
`m=6`

therefore

The system has no solution

Part 5) we have

`x-2y=6` ------> equation A

`3x-6y=18` ------> equation B

Multiply the equation A by
`3`

`3(x-2y)=6*3` ----->
`3x-6y=18`

The equation A and the equation B are the same

therefore

The system has infinitely solutions

Part 6) we have

`y-7x=-14` ------> equation A

`7y-49x=-2` ------> equation B

Multiply the equation A by
`7`

`7(y-7x)=-14*7` ----->
`7y-49x=-98`

Equation A and equation B are parallel lines, because has the same slope

the slope is equal to
`m=7`

therefore

The system has no solution