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6.5 Intersections

So far we haven't seen how our receiver gets tackled. On the football field, many players run lines on the field.  As any linebacker will tell you, sometimes the lines intersect.  Graphs, like football, are much more interesting when the lines intersect.

Example 6.7:  Tony catches the ball on the opponent's 46-yard line.  He still runs at 8 yards per second.  The defensive back covering Tony, Eric, has been "burned" (he let he receiver get past him) and is at the 50 yard line when Tony catches the ball.  Eric can run at 9 yards per second.  If no other players are involved in the pursuit, will Eric catch up to Tony before the goal line?  If so, where does he catch him?

Solution:  The first thing we notice from the problem is that the defensive back, Eric, can run faster than the receiver, Tony.  So we know that if Eric has enough time, he can catch Tony.  We start out by writing the equation for each player's location.

For Tony:      y  = -8x + 46          where  y = field position, in yards

                                                             x = time elapsed since the catch, in seconds

For Eric:         y  = -9x + 50              

The key to solving this problem is knowing that when the players intersect, they will be at the same place at the same time.  In other words, Eric's location will be the same as Tony's location. 

            Therefore:                 y  = -8x + 46 = -9x +50 

                                    -8x + 46 = -9x +50

This is a problem we can solve using our standard algebraic laws. 

Since we now know when they will intersect, we can determine where they will intersect.  We can plug the 4 seconds into either Tony's or Eric's equation to find the position where they meet.

Itπs always a good idea to double-check your answer by plugging the 4 into the other equation. 

Eric catches up with Tony at the 14 yard line.

If we graph the two lines, we see that the lines intersect at 4 seconds and 14 yards. 

What would have happened if Eric and Tony ran at the same speed?  Eric would have never caught up with Tony.  If their speeds were the same, their m values or slopes in the equation y = mx + b would be the same. We know that if they ran at the same speed, they would stay the same distance apart down the field.  If we graph this, we see that the distance between them remains constant as they run downfield.  We also notice that the two lines are parallel.

Any lines which have the same slope will be parallel.

Example 6.8:  Dave is a wide receiver who can run 9 yd./sec and is being covered by Phil who can run 8 yd./sec. A pass play starts at the offense's 6 yard line (yards increasing).  Dave catches the pass at the offense's 42 yard line while Phil is at the 39 yard line.  Where do the two players intersect?

Solution: As before, we set up a system of equations:

For Dave       y  = 9x + 42  where y = field position, in yards

                                                            x = time elapsed since the catch, in seconds

For Phil          y = 8x + 39

Again, their positions will be equal when they pass each other.  In other words, their y-values will be the same.  So, 

The players passed each other at -3 seconds. 

What does it mean to say that they passed each other at -3 seconds?  The zero point of the time was when Dave caught the ball.  If x = -3, they must have passed each other three seconds before Dave caught the ball.  We could have suspected this by first looking at the problem. Since Dave was faster and was farther down the field, the only time they could have passed each other was when Dave passed Phil earlier in the play.  If we solve one of the equations for y at x = -3, we can see that they did pass each other earlier down the field. 

The players passed each other at the 15 yard line.  The graph of the equations is shown in Figure 6.13.

In this problem, the intersection of the line occurs in the second quadrant.

Equations that use the same variables are called simultaneous equations.  Problems involving simultaneous equations can only be solved if the number of equations equals the number of variables in the problem.  In the problems we've done so far, we've had two equations and two variables or unknowns. 

Even when the number of equations equals the number of variables, a solution may not exist.  In the problem where the receiver and the defensive back ran at the same speed, their lines on the graph never intersected.  There was no solution.  Two simultaneous equations will have no solution if the slopes are the same.  Equations with the same slope are said to be dependent. When the slopes of two simultaneous equations are different, the equations are said to be independent.  The solution to two independent equations is shown by the intersection of their lines on a graph.