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Question

Part A: Explain why the x-coordinates of the points where the graphs of the equations y = 4–x and y = 2x + 3 intersect are the solutions of the equation 4–x = 2x + 3. (4 points)

Part B: Make tables to find the solution to 4–x = 2x + 3. Take the integer values of x between −3 and 3. (4 points)

Part C: How can you solve the equation 4–x = 2x + 3 graphically? (2 points)

Answer 1

Par A)

 

The x-coordinates are solutions because if you plug in the same x value to evaluate both equations, you will get the same y-coordinate.

 

 

Part B)

 

Make the table of values using the given domain.  Evaluate both functions for each of these x values.  The value of x that make the y values of BOTH equations the same is the solution.

 

x       |       (4 - x)      |      (2x + 3)

_______________________________

 

-3                  7                      -3

-2                  6                      -1

-1                  5                       1

0                   4                       3

1                   3                       5

2                   2                       7

3                   1                       9

 

As you can see, none of these x values within the domain have y values that are the same.  So we know the solution is none of these integers.

 

 

Part C)

 

To graph these functions, we draw the lines that represent each equation on the same coordinate system.  To do this, we plot points, we use the slope and y-intercept.  We use the y-intercept as the first point.  Then we plot the second point, with the slope and y-intercept as the base point.

 

For   y = 4 - x

 

The first point is (0, 4).

 

The second point is (1, 3).

 

Connect these two points throughout the entire coordinate system.

 

 

For    y = 2x + 3

 

The first point is (0, 3).

 

The second point is (1, 5).

 

 

Connect these two points throughout the entire coordinate system.

 

Now that you have your two lines, look for the point of intersection.