The relationships between x, the number of weeks, and y, the number of miles Ballio and Mary Beth bike each week, are graphed on the coordinate plane below. What is the point of intersection?
Ballio
Mary Beth
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Time (weeks)
Number of miles
The point of intersection is (
,
).
Good work!
Now that you have found the point of intersection, (2, 80), let’s check that work algebraically. The equations for the relationship between weeks and miles for Ballio and Mary Beth are shown below. Since the number of miles Ballio bikes will be equal to the number of miles Mary Beth bikes at the solution to this system, you can set the expression for the number of miles they each bike, 50+15x and 60+10x, equal to one another. Do that now.
Ballio
Mary Beth
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Time (weeks)
Number of miles
50+15x
=
60+10x
Nice!
Let’s solve the equation for x, the number of weeks when Ballio and Mary Beth bike the same number of miles, to confirm that it’s the same as the x-coordinate of the point of intersection. Since there are x terms on both sides of the equation, let’s get rid of the x term from one of the sides. What move can you make to get rid of the x term on the right side of the equation, 10x, while keeping the equation balanced?
Ballio’s miles = Mary Beth's miles
50+15x = 60+10x
Multiply each side by 10x
Add 10x to each side
Subtract 10x from each side
Excellent!
We can subtract 10x on both sides to get rid of the 10x on the right while keeping the equation balanced. Record subtracting 10x on both sides of the equation.
Ballio's miles
=
Mary Beth's miles
50+15x
=
60+10x
−
10x
−
10x
Nice!
Subtract 10x on both sides and rewrite the equation to represent the remaining values.
Ballio's miles
=
Mary Beth's miles
50+15x
=
60+10x
−
10x
−
10x
=
Great job!
We have all of the variables on the left. Now, keep going. What move can you make next to get rid of 50 on the left so that the x term, 5x, is alone on the left?
Ballio's miles
=
Mary Beth's miles
50+15x
=
60+10x
−
10x
−
10x
50+5x
=
60
Add 50 to both sides
Subtract 50 from both sides
Add 5x on both sides.
Good work!
You can subtract 50 on both sides so that the x term remains alone on the left, and the equation is balanced. Record that in the equation.
Ballio's miles
=
Mary Beth's miles
50+15x
=
60+10x
−
10x
−
10x
50 + 5x
=
60
−
50
−
50
Excellent!
Subtract 50 from both sides and rewrite the equation to represent the remaining values.
Ballio's miles
=
Mary Beth's miles
50+15x
=
60+10x
−
10x
−
10x
50 + 5x
=
60
−
50
−
50
=
You got it!
Keep going. Make a balanced move so that x is alone on the left, and then rewrite the equation to solve for the value of x.
Ballio's miles
=
Mary Beth's miles
50+15x
=
60+10x
−
10x
−
10x
50 + 5x
=
60
−
50
−
50
5x
=
10
x
=
You got it!
You just found that x = 2. That’s the same as the x-coordinate at the point of intersection. Let’s see if the corresponding y-coordinate is the same, too. What’s the value of y when x = 2? You can use either equation to find out. You can solve on paper, and then enter your answer on Zearn.
Miles Ballio bikes per week: y= 50+15x
Miles Mary Beth bikes per week: y= 60+10x
y= 50+15x
y= 60+10x
When x= 2, y=
Nice!
You found that the point of intersection on the graph is (2, 80), and you solved algebraically to find that x = 2 and y = 80. What does the point (2, 80) mean in the context of this problem?
Ballio
Mary Beth
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Time (weeks)
Number of miles
Ballio and Mary Beth both bike 80 miles per week in week 2.
Ballio bikes 80 miles and Mary Beth bikes 2 miles.
Ballio and Mary Beth both bike 2 miles per week in week 80.
1 answer
The point (2, 80) means that after 2 weeks, both Ballio and Mary Beth will be biking the same distance of 80 miles per week.
So the correct interpretation of the point (2, 80) in the context of this problem is:
Ballio and Mary Beth both bike 80 miles per week in week 2.
