The relationships between x, the number of weeks, and y, the number of miles Dylan and Nora bike each week, are graphed on the coordinate plane below. What is the point of intersection?
Dylan
Nora
0
1
2
3
4
5
6
7
8
9
10
0
20
40
60
80
100
120
140
160
180
200
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 Dylan and Nora are shown below. Since the number of miles Dylan bikes will be equal to the number of miles Nora bikes at the solution to this system, you can set the expression for the number of miles they each bike, 60+10x and 120−20x, equal to one another. Do that now.
Dylan
Nora
0
1
2
3
4
5
6
7
8
9
10
0
20
40
60
80
100
120
140
160
180
200
Time (weeks)
Number of miles
60+10x
=
120−20x
Excellent!
Let’s solve the equation for x, the number of weeks when Dylan and Nora 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, 20x, while keeping the equation balanced?
Dylan’s miles = Nora's miles
60+10x = 120−20x
Multiply each side by 20x
Add 20x to each side
Subtract 20x from each side
Good work!
We can add 20x on both sides to get rid of the 20x on the right while keeping the equation balanced. Record adding 20x on both sides of the equation.
Dylan's miles
=
Nora's miles
60+10x
=
120−20x
+
20x
+
20x
Excellent!
Add 20x on both sides and rewrite the equation to represent the remaining values.
Dylan's miles
=
Nora's miles
60+10x
=
120−20x
+
20x
+
20x
=
Great job!
We have all of the variables on the left. Now, keep going. What move can you make next to get rid of 60 on the left so that the x term, 30x, is alone on the left?
Dylan's miles
=
Nora's miles
60+10x
=
120−20x
+
20x
+
20x
60+30x
=
120
Subtract 60 from both sides
Add 30x on both sides.
Add 60 to both sides
Excellent!
You can subtract 60 on both sides so that the x term remains alone on the left, and the equation is balanced. Record that in the equation.
Dylan's miles
=
Nora's miles
60+10x
=
120−20x
+
20x
+
20x
60 + 30x
=
120
−
60
−
60
Nice!
Subtract 60 from both sides and rewrite the equation to represent the remaining values.
Dylan's miles
=
Nora's miles
60+10x
=
120−20x
+
20x
+
20x
60 + 30x
=
120
−
60
−
60
=
Nice!
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.
Dylan's miles
=
Nora's miles
60+10x
=
120−20x
+
20x
+
20x
60 + 30x
=
120
−
60
−
60
30x
=
60
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 Dylan bikes per week: y= 60+10x
Miles Nora bikes per week: y= 120−20x
y= 60+10x
y= 120−20x
When x= 2, y=
1 answer
To find the value of \( y \) when \( x = 2 \), you can use either of the equations provided.
Using Dylan's equation: \[ y = 60 + 10x \] Substituting \( x = 2 \): \[ y = 60 + 10(2) = 60 + 20 = 80 \]
Using Nora's equation: \[ y = 120 - 20x \] Substituting \( x = 2 \): \[ y = 120 - 20(2) = 120 - 40 = 80 \]
In both cases, when \( x = 2 \), \( y = 80 \).
Thus, when \( x = 2 \), \( y = 80 \).
The point where Dylan and Nora bike the same number of miles each week is \( (2, 80) \).
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