Question

Consider the system of linear equations.

2y = x + 10

3y = 3x + 15






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


The system has one solution.


The system graphs parallel lines.


Both lines have the same slope.


Both lines have the same y-intercept.


The equations graph the same line.


The solution is the intersection of the 2 lines

Answer 1

2y = x + 10
3y = 3x + 15
multiply the first by 3
6y=3x+10
6y-10=3x
put 6y-10 in place of 3x in the second one
3y=6y-10+15
-3y=5
y=-5/3
x=-20/3
these statements are true based on answer
The system has one solution.
Both lines have the same y-intercept.
The solution is the intersection of the 2 lines

Answer 2

Statement 1 : True

Statement 2: False

Statement 3: False

Statement 4:True

Statement 5:False

Statement 6:True

Explanation:

Given :
`2y = x + 10`

`3y = 3x + 15`

To Find: Which statements about the system are true?

Solution:

Slope intercept form:
`y =mx+c`

where m is the slope

Convert the given equations in slope intercept form

Equation 1:
`2y = x + 10`

`y = (x)/(2)+(10)/(2)`

`y = (x)/(2) +5` --A

On comparing with general form

Slope = m =
`(1)/(2)`

Equation 2:
`3y = 3x + 15`

`y = (3)/(3)x + (15)/(3)`

`y = x +5` --B

On comparing with general form

Slope = m = 1

Substitute the value of x form B in A

`y = (y-5)/(2) +5`

`y-5 = (y-5)/(2)`

`2y-10 = y-5`

`y=5`

Substitute the value of y in B to get value of x

`5 = x +5`

`x=0`

Thus the solution is (0,5)

Thus The system has one solution.

So, Statement 1 is true

Since the slopes of the given lines are not same so the lines are not parallel

Thus the statement 2 is false.

Slope of line 1 is
`(1)/(2)` and slope of line 2 is 1

Since the slopes are not same

So, statement 3 is false.

Equation 1:
`2y = x + 10`

To find y intercept substitute x =0

`2y = 0 + 10`

`y =(10)/(2)`

`y =5`

Equation 2:
`3y = 3x + 15`

To find y intercept substitute x =0

`3y = 3(0) + 15`

`y =(15)/(3)`

`y =5`

So,statement 4 is true Both lines have the same y-intercept.

Statement 5: The equations graph the same line.

Statement 5 is false (refer the attached graph)

Statement 6 is true The solution is the intersection of the 2 lines (refer the attached graph)