Part a. explain why the x-coordinates of the points where the graphs of the equations y = 2−x and y = 4x + 3 intersect are the solutions of the equation 2−x = 4x + 3. part b. make tables to find the solution to 2−x = 4x + 3. take the integer values of x only between −3 and 3. part c. how can you solve the equation 2−x = 4x + 3 graphically?

y = 2−x and y = 4x + 3

Let I be the point of intersection:

I belongs to the line y=2-x and at the same time I belongs to y=4x+3, then the coordinates of I are the same for y =2-x & y= 4x+2, in short if we replace the coordinates in y = 2-x & in y=4x+3 by their respective values, we will find an equality.

b) 2−x = 4x + 3

Replace x  in  the equation 2−x = 4x + 3  with the here below values to find if an equality exists

for x = -3 then  2-(-3) = 4(-3)+3 → 5 = -9 IMPOSSIBE, it's not an equality

for x = 2 then  2-2 = 4(2)+3 → 0 = 11 IMPOSSIBE, it's not an equality

for x = 1 then  2-(1) = 4(1)+3 → 1 = 7 IMPOSSIBE, it's not an equality

for x = - 1 then  2-(-1) = 4(-1)+3 → 3 = -1 IMPOSSIBE, it's not an equality

and you can replace x with all integers from - 3 to + 3 and you will find an INEQUALITY, so all these values are not a solution of thev equation

c. Solving the equation 2−x = 4x + 3 
2−x = 4x + 3  

1st add x to both sides: 2-x + x = 4x + 3 + x →2 = 5x + 3

2nd to this new equation 2 = 5x + 3, subtract 3 from both sides:

2 - 3= 5x + 3 - 3 →-1 = 5x

3rd in this new equation -1=5x, divide both sides by 5 → -1/5 = x
 And x = -1/5 is the solution of the system. To find the y value, you replace in any of the 2 equation  y = 2−x and y = 4x + 3, x by its value (-1/5) and you will find y = 11/5