a Examining the system, we see that in the second equation, x is already isolated. Therefore, its most easily solved by using the Substitution Method.
2x+6y=10 x=8-3y
The second equation is already isolated for the variable x. Therefore, we can substitute the expression for x in the first equation and solve for y.
We have reached a contradiction as 16 does not equal 10. This means the system has not solutions.
b To graph a linear function, we need at least two points that fall on its graph. Let's rewrite the equations so that they arein slope-intercept form.
2x+6y=10 x=8-3y ⇔ y=- 13x+ 53 y=- 13x+ 83
Next, we will choose two x-values and calculate their corresponding y-values for each function.
|c|c|c|
[-1em]
x & - 1/3x+5/3 & y [0.8em]
[-1em]
2 & -1/3( 2)+5/3 & 5 [0.8em]
[-1em]
5 & -1/3( 5)+5/3 & 0 [0.8em]
|c|c|c|
[-1em]
x & - 1/3x+8/3 & y [0.8em]
[-1em]
5 & -1/3( 5)+8/3 & 1 [0.8em]
[-1em]
8 & -1/3( 8)+8/3 & 0 [0.8em]
By plotting each set of points in a coordinate plane, we can draw their graphs.
c From the diagram and slope-intercept form, we see that the functions have the same slope. Since they also have different y-intercepts, they must be parallel lines.
