First of all, a solution to a system can be thought of in two ways: Graphically, the solution is where the lines produced by the two equations intersect. If you graphed y = -x +3 and also graphed y = 2x + 1, the solution is where the graphs 'cross' or intersect. Numerically, the solution to the system is where a particular x-value produces the same y-value in each equation. The solution would be found numerically when we plugged in a particular value for x into both equations and the y-value is the same for both equations.
For part 5a.) We know the solution is between the highlighted rows of x = 0.5 and x = 1 because the line y = -x +3 decreases while the line y = 2x+1 increases. It is easiest to think of it graphically. We know that y = -x +3 and y = 2x+1 are both lines, so we can easily sketch the graphs of each using the table. The line y = -x+3 would go through the point (0.5, 2.5) and then continue downward to the point (1,2). Meanwhile the line y = 2x + 1 would go through the point (0.5,2) and continue upward to the point (1,3). Thus in-between the x-values 0.5 and 1, one of the lines is going downward from 2.5 to 2, while the other is going upward from 2 to 3. Somewhere in this range the two lines must have an intersection point, which we know is the solution to the system.
For part 5b.) we can complete the table by plugging in the given x-value into each of the equations. The table should look as follows:
x y = -x+3 y=2x+1
0.5 2.5 2
0.6 2.4 2.2
0.7 2.3 2.4
0.8 2.2 2.6
0.9 2.1 2.8
1 2 3
