a The slope-intercept form of a line is y=mx+b, where m is the slope and b is the y-intercept. Use the information provided to obtain the slope and y-intercept of both lines.
y= mx+ b
In this form, m represents the slope and b the y-intercept. Let x be the time, in hours, that has passed after we and our friend started hiking. Let y be the distance from the trailhead.
Writing the System
Since we start at the trailhead, we know that y=0 when x=0. Therefore, the y-intercept of the line that models our walk is 0. Moreover, we walk 5 miles per hour. This means the slope of our line is 5.
y= 5x+ 0 ⇔ y=5x
Our friend starts hiking 3 miles away from the trailhead, and he walks 3 miles per hour. Following the same reasoning as before, the y-intercept of his walk is 3 and the slope is also 3.
y= 3x+ 3
Combining the equations together forms a system of linear equations.
y=5x y=3x+3
Graphing the System
To draw the graph of a line, we plot the y-intercept and use the slope to obtain another point on the line. Then, we connect these points using a straight edge. Let's draw the graph of y=5x.
We will follow the same procedure to draw the graph of y=3x+3.
b We will use our graph to answer this question. Let's pay close attention to the point of intersection of the lines.
We can see that the lines intersect when x=1.5, and not when x=1. This means that we will be in the same location as our friend after 1.5 hours, and not after one hour. Therefore, our friend is incorrect.
