Question

Which statements about the system are true? Check all that apply.

y = 1/3x – 4
3y – x = –7
A- The system has one solution.
B- The system consists of parallel lines.
C- Both lines have the same slope.
D- Both lines have the same y–intercept.
E- The equations represent the same line.
F- The lines intersect.

Answer 1

we have

`y=(1)/(3)x-4` ------> equation A

`3y-x=-7`

isolate the variable y

`3y=x-7`

`y=(1)/(3)x-(7)/(3)` -------> equation B

Let's verify each of the statements

Statements

case A) The system has one solution

The statement is false

The lines are parallel, because their slopes are equal

the slope is equal to
`m=(1)/(3)`

therefore

The system has no solution

case B) The system consists of parallel lines

The statement is true

Because, the slopes of the lines are the same

case C) Both lines have the same slope

The statement is true

The slope both lines is equal to
`m=(1)/(3)`

case D) Both lines have the same y–intercept

The statement is false

we know that

The y-intercept is the value of y when the value of x is equal to zero

For
`x=0`

Find the y-intercept of the equation A

`y=(1)/(3)*0-4=-4`

Find the y-intercept of the equation B

`y=(1)/(3)*0-(7)/(3)=-(7)/(3)`

so

`-4\\eq -(7)/(3)`

case E) The equations represent the same line

The statement is false

The equations represent different lines, because both lines have different y-intercept

case F) The lines intersect.

The statement is false

Because, since the lines are parallel so they never intersects

therefore

the answer is

B- The system consists of parallel lines

C- Both lines have the same slope

Answer 2

The given lines are y = 1/3x - 4 and y = 1/3x -7/3

Lets go through each of the option

A - The two lines are parallel, hence they don't have any common solution.

B - Yes, this is true. Both the lines are parallel. As both the lines have same slope.

C - Yes, both the lines have same slope which is equal to 1/3

D - No, both the lines have different y intercepts . Y intercept of the line y = 1/3x - 4 is -4 and Y intercept of the line y=1/3x -7/3 is -7/3

E - No, the equation represents different lines.

F - Since the lines are parallel so they never intersects.

Correct Ones are B, C, F

Hope this helps ..!!