Question

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

y = x – 4
3y – x = –7
The system has one solution.
The system consists of parallel lines.
Both lines have the same slope.
Both lines have the same y–intercept.
The equations represent the same line.
The lines intersect.

Answer 1

Answer and explanation:

Given : Equations
`y=x-4` and
`3y-x=-7`

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

Solution :

First we solve the system of equations,

`y=x-4` .....(1)

`3y-x=-7` ......(2)

Substitute y from (1) in (2),

`3(x-4)-x=-7`

`3x-12-x=-7`

`2x=5`

`x=(5)/(2)`

Substitute the value of x in (1),

`y=(5)/(2)-4`

`y=(5-8)/(2)`

`y=(-3)/(2)`

1) The system has one solution i.e.
`((5)/(2),-(3)/(2))`

2) The system has solution which means it is not parallel lines.

Writing equation in slope from,
`y=mx+c`

`y=x-4` where m=1 and c=-4

`3y-x=-7` where
`m=(1)/(3)` and c=-4

3) Both lines have different slopes.

4) Both lines have the same y-intercept.

5) The equations represent the different lines.

6) The lines intersect at
`((5)/(2),-(3)/(2))`

Answer 2

The correct answers are:

• The system has one solution.

• The lines intersect.

Explanation:

We are given system of linear equations as:

y=x-4--------(1)

and 3 y-x= -7----------(2)

on substituting the value of 'y' from equation (1) into equation (2) we have:

3(x-4)-x=-7

⇒ 3x-3×4-x=-7

⇒ 3x-x= -7+12

⇒ 2x=5

⇒
`x=(5)/(2)`

also putting the value of x into equation (1) we have:

`y=(5)/(2)-4=(5-8)/(2)=(-3)/(2)`

• Hence, we get a unique value of x and y on solving the system of linear equations.

Hence, the system has one solution.

• the slope for line y=x-4 is 1 ( since on comparing the equation with y=mx+c; where m denotes the slope of line and c is the y-intercept)

y-intercept is -4

but the slope of line 3y-x = -7 i.e.

`y=(x-7)/(3)=(x)/(3)-(7)/(3)`

hence the slope of second line is:
`(1)/(3)`.

and y-intercept is
`(-7)/(3)`.

• They represent different lines.

• the lines intersect at the point
  `((5)/(2),(-3)/(2))`