Answer:
Statement 1 : True
Statement 2: False
Statement 3: False
Statement 4:True
Statement 5:False
Statement 6:True
Step-by-step explanation:
Given : [tex]2y = x + 10[/tex]
[tex]3y = 3x + 15[/tex]
To Find: Which statements about the system are true?
Solution:
Slope intercept form: [tex]y =mx+c[/tex]
where m is the slope
Convert the given equations in slope intercept form
Equation 1: [tex]2y = x + 10[/tex]
[tex]y = \frac{x}{2}+\frac{10}{2}[/tex]
[tex]y = \frac{x}{2} +5[/tex] --A
On comparing with general form
Slope = m = [tex]\frac{1}{2}[/tex]
Equation 2: [tex]3y = 3x + 15[/tex]
[tex]y = \frac{3}{3}x + \frac{15}{3}[/tex]
[tex]y = x +5[/tex] --B
On comparing with general form
Slope = m = 1
Substitute the value of x form B in A
[tex]y = \frac{y-5}{2} +5[/tex]
[tex]y-5 = \frac{y-5}{2}[/tex]
[tex]2y-10 = y-5[/tex]
[tex]y=5[/tex]
Substitute the value of y in B to get value of x
[tex]5 = x +5[/tex]
[tex]x=0[/tex]
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 [tex]\frac{1}{2}[/tex] and slope of line 2 is 1
Since the slopes are not same
So, statement 3 is false.
Equation 1: [tex]2y = x + 10[/tex]
To find y intercept substitute x =0
[tex]2y = 0 + 10[/tex]
[tex]y =\frac{10}{2}[/tex]
[tex]y =5[/tex]
Equation 2: [tex]3y = 3x + 15[/tex]
To find y intercept substitute x =0
[tex]3y = 3(0) + 15[/tex]
[tex]y =\frac{15}{3}[/tex]
[tex]y =5[/tex]
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)
