Respuesta :
Answer:
The correct answers are:
- The system has one solution.
- The lines intersect.
Step-by-step 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
⇒ [tex]x=\dfrac{5}{2}[/tex]
also putting the value of x into equation (1) we have:
[tex]y=\dfrac{5}{2}-4=\dfrac{5-8}{2}=\dfrac{-3}{2}[/tex]
- 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.
[tex]y=\dfrac{x-7}{3}=\dfrac{x}{3}-\dfrac{7}{3}[/tex]
hence the slope of second line is: [tex]\dfrac{1}{3}[/tex].
and y-intercept is [tex]\dfrac{-7}{3}[/tex].
- They represent different lines.
- the lines intersect at the point [tex](\dfrac{5}{2},\dfrac{-3}{2})[/tex]
Answer and explanation:
Given : Equations [tex]y=x-4[/tex] and [tex]3y-x=-7[/tex]
To find : Which statements about the system are true? Check all that apply.
Solution :
First we solve the system of equations,
[tex]y=x-4[/tex] .....(1)
[tex]3y-x=-7[/tex] ......(2)
Substitute y from (1) in (2),
[tex]3(x-4)-x=-7[/tex]
[tex]3x-12-x=-7[/tex]
[tex]2x=5[/tex]
[tex]x=\frac{5}{2}[/tex]
Substitute the value of x in (1),
[tex]y=\frac{5}{2}-4[/tex]
[tex]y=\frac{5-8}{2}[/tex]
[tex]y=\frac{-3}{2}[/tex]
1) The system has one solution i.e. [tex](\frac{5}{2},-\frac{3}{2})[/tex]
2) The system has solution which means it is not parallel lines.
Writing equation in slope from, [tex]y=mx+c[/tex]
[tex]y=x-4[/tex] where m=1 and c=-4
[tex]3y-x=-7[/tex] where [tex]m=\frac{1}{3}[/tex] 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 [tex](\frac{5}{2},-\frac{3}{2})[/tex]
