Question

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. (4 points)
Part B: Make tables to find the solution to 2−x = 4x + 3. Take the integer values of x only between −3 and 3. (4 points)
Part C: How can you solve the equation 2−x = 4x + 3 graphically? (2 points)

Answer 1

Answer:

Step-by-step explanation:

Part A:

The solution of a system is not just the x coordinates; it is the whole coordinate pair that is the solution, where both x and y are the same.  Normally, when you have a system and are solving them simultaneously, you are looking for the point at which they are equal.  This is a very useful concept in business and finance, both in the home for personal information, and in the office setting where companies are.  Where the 2 equations intersect is a point where they are equal.  

Part B:

The graphs do not intersect right at a perfect integer of x.  Therefore, we will solve these equations simultaneously to solve first for x, then we will plug in x to solve for y.  Since we have the equations set to equal each other, we can solve for x by getting everything on one side of the equation and then setting it equal to 0.  

2 - x = 4x + 3 so

5x + 1 = 0.  Solving for x,

5x = -1 so

$x=-\frac{1}{5}$

The y coordinate can be found by subbing in this value of x into either equation.  If y = 2 - x, and x = -1/5, then

y = 2 -(-1/5) and y = 2 + 1/5 and y = 10/5 + 1/5 gives us that y = 11/5

Thus, the coordinate pair that is the solution to that system is

$(-\frac{1}{5},\frac{11}{5})$

Part C:

You would solve the system graphically by graphing both lines on the same window.  However, since their intersection is not an integer pair, but are fractions, you would not be able to tell EXACTLY where they intersect.  From the graphing window, you would hit your 2nd button then "trace" which is in the row at the very top of the buttons below the window.  Then hit 5:  intersect.  You'll be back to your graph of the lines, and there will be a cursor blinking along the line you graphed under Y1.  Move the cursor til it is right over the intersection of the lines and hit "enter".  Then you'll be back to the graphs with a blinking cursor over the line you entered in Y2.  Move that cursor along the line til it is dead-center over the other point on the first line and hit "enter" again.  At the bottom, you will see the x and y coordinates that are the intersection of this system.