Question

Part A: Explain why the x-coordinates of the points where the graphs of the equations y = 4−x and y = 2−x+1 intersect are the solutions of the equation 4−x = 2−x+1. (4 points) Part B: Make tables to find the solution to 4−x = 2−x+1. Take the integer values of x between −2 and 2. (4 points) Part C: How can you solve the equation 4−x = 2−x+1 graphically? (2 points)

Answer 1

A.  We have two lines:  y = 2-x   and   y = 4x+3 Given two simultaneous equations that are both required to be true.. the solution is the points where the lines cross...  Which is where the two equations are equal..  Thus the solution that works for both equations is when 2-x = 4x+3 because where that is true is where the two lines will cross and that is the common point that satisfies both equations.   B.  2-x = 4x+3   x     2-x      4x+3
______________

-3     5       -9
-2     4       -5
-1     3       -1
 0     2        3
 1     1        7
 2     0      11
 3    -1      15

The table shows that none of the integers from [-3,3] work because in no case does
2-x = 4x+3   To find the solution we need to rearrange the equation to the form x=n 2-x = 4x+3 2 -x + x = 4x + x +3 2 = 5x + 3 2-3 = 5x +3-3 5x = -1 x = -1/5   The only point that satisfies both equations is where x = -1/5 Find y:   y = 2-x  = 2 - (-1/5) = 2 + 1/5 = 10/5 + 1/5 = 11/5 Verify we get the same in the other equation y = 4x + 3   =  4(-1/5) + 3 = -4/5 + 15/5 = 11/5   Thus the only actual solution, being the point where the lines cross, is the point (-1/5, 11/5)   C.  To solve graphically  2-x=4x+3 we would graph both lines...   y = 2-x  and   y = 4x+3 The point on the graph where the lines cross is the solution to the system of equations ... [It should be, as shown above, the point (-1/5, 11/5)]   To graph y = 2-x   make a table.... We have already done this in part B   x     2-x                x    4x+3 _______              ________ -1    3                   -1      -1  0    2                    0       3  1    1                    1       7   Just graph the points on a cartesian coordinate system and draw the two lines.  The solution is, as stated, the point where the two lines cross on the graph.  

Hope this helps.

Answer 2

You probably mean the equations are y = 4 - x and y = 2x -1. Else with given equations, the solution is not possible. The steps of the solution are listed below:

Part A) 
The solution to the equation 4 - x = 2x + 1 are the points where both the equations have same value for a given value of x. This means, if a points (x,y) lies on both the lines it will be the solution of the given equations. Since that point lies on both the lines, both lines will cross that point. Crossing that point can be seen as intersection of two lines at that point. 
Therefore, x-coordinates of the points where the graphs of equations y = 4 -  x and y = 2x + 1 intersect are the solutions of the equation 4 - x = 2x + 1

Part B)
The table is attached in the images below. As seen from the table, both lines have same value of y for x =1. So x=1 is the solution to the equation 4 - x = 2x + 1

Part C)
In order to solve the equation 4 - x = 2x + 1 , we plot the individual lines y = 4 - x and y = 2x + 1 on the graph. The point of intersection of two lines will be the solution to the equation. The graph of lines is attached below.