Zara and Steven both track the number of miles they bike each week. Zara was already biking 60 miles per week, and started biking 20 more miles each week. Steven was already biking 90 miles per week, and started biking 10 more 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 Zara and Steven bike each week, are graphed on the coordinate plane below. What is the point of intersection?
Zara
Steven
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 (
,
).
Great job!
Now that you have found the point of intersection, (3, 120), let’s check that work algebraically. The equations for the relationship between weeks and miles for Zara and Steven are shown below. Since the number of miles Zara bikes will be equal to the number of miles Steven bikes at the solution to this system, you can set the expression for the number of miles they each bike, 60+20x and 90+10x, equal to one another. Do that now.
Zara
Steven
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+20x
=
90+10x
Good work!
Let’s solve the equation for x, the number of weeks when Zara and Steven 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?
Zara’s miles = Steven's miles
60+20x = 90+10x

Subtract 10x from each side

Multiply each side by 10x

Add 10x to each side
Nice!
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.
Zara's miles
=
Steven's miles

60+20x
=
90+10x
−

10x
−

10x
Excellent!
Subtract 10x on both sides and rewrite the equation to represent the remaining values.
Zara's miles
=
Steven's miles

60+20x
=
90+10x
−
10x
−
10x

=

Answer 1

Once you subtract \(10x\) from both sides of the equation, you have:

\[ 60 + 20x = 90 + 10x - 10x \]

This simplifies to:

\[ 60 + 20x = 90 \]

Now, you can further solve for \(x\) by isolating the variable.

Next, subtract 60 from both sides of the equation:

\[ 60 + 20x - 60 = 90 - 60 \]

This simplifies to:

\[ 20x = 30 \]

Now, divide both sides by 20 to solve for \(x\):

\[ x = \frac{30}{20} = \frac{3}{2} = 1.5 \]

However, this does not match the previously mentioned point of intersection (3, 120). Let's re-evaluate the equation:

Originally, you should have:

1. \(60 + 20x = 90 + 10x\)

After subtracting \(10x\) from both sides, you get:

\[ 60 + 20x - 10x = 90 \]

This simplifies to:

\[ 60 + 10x = 90 \]

Next, subtract 60 from both sides:

\[ 10x = 90 - 60 \]

Now, this simplifies to:

\[ 10x = 30 \]

Finally, divide each side by 10:

\[ x = \frac{30}{10} = 3 \]

So, when \(x = 3\), they both bike the same number of miles.

Now, let's find the number of miles they bike after 3 weeks to confirm the y-coordinate:

For Zara:

\[ y = 60 + 20(3) = 60 + 60 = 120 \]

For Steven:

\[ y = 90 + 10(3) = 90 + 30 = 120 \]

Thus, the point of intersection is indeed \((3, 120)\), confirming your earlier conclusion.