Exercise 108 Page 506

Will The Graphs Intersect?

Since both graphs are lines, we will be able to write them in slope-intercept form. If two lines in a coordinate plane do not intersect, they are parallel lines. This means they have the same slope but different $y\text{-}$intercepts. Examining the diagram we see that they have different slopes, one negative and one positive. Therefore, we can conclude that they will intersect.

Finding The Equations

To find the equations, we have to identify their $y\text{-}$intercept and slope. Examining the diagram, we see that one graph intersects the $y\text{-}$axis at $(0,10)$ and the other intersect the $y\text{-}$axis at $(0,3).$

Next we will determine the slope. By locating a second lattice point through which the graphs pass, we can determine the slope by dividing the number of steps in the vertical direction with the number of steps in the positive horizontal direction between the points.

If the lines intersect, we should be able to equate the function's right-hand sides and get a solution. Let's try to do that. As we can see, the lines intersect at $x=12.$