Dylan and Nora both track the number of miles they bike each week. Dylan was already biking 60 miles per week, and started biking 10 more miles each week. Nora was already biking 120 miles per week, and started biking 20 less miles each week. Is there a time when they bike the same number of miles each week?

Question

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

Answer 1

Great progress with the algebra! To isolate \(x\) if we have:

\[ 60 + 30x = 120 \]

The next step is to subtract \(60\) from both sides of the equation to get rid of the \(60\) on the left:

\[ 60 + 30x - 60 = 120 - 60 \]

This simplifies to:

\[ 30x = 60 \]

Now, to solve for \(x\), we can divide both sides by \(30\):

\[ x = \frac{60}{30} \]

This simplifies to:

\[ x = 2 \]

So, Dylan and Nora bike the same number of miles after \(2\) weeks. Now, let's find the corresponding number of miles. We can substitute \(x\) back into either equation. Let's use Dylan's equation:

\[ y = 60 + 10x \] Substituting \(x = 2\):

\[ y = 60 + 10(2) = 60 + 20 = 80 \]

So, the point of intersection is indeed \((2, 80)\). This means that after \(2\) weeks, both Dylan and Nora will be biking \(80\) miles per week.

The final answer is: The point of intersection is \( (2, 80) \).