The relationships between x, the number of weeks, and y, the number of miles Jenna and Luna run each week, are graphed on the coordinate plane below. What is the point of intersection?
Jenna
Luna
0
1
2
3
4
5
6
7
8
9
10
0
8
16
24
32
40
48
56
64
72
80
Time (weeks)
Number of miles
The point of intersection is (
,
).
Good work!
Now that you have found the point of intersection, (5, 40), let’s check that work algebraically. The equations for the relationship between weeks and miles for Jenna and Luna are shown below. Since the number of miles Jenna runs will be equal to the number of miles Luna runs at the solution to this system, you can set the expression for the number of miles they each run, 20+4x and 30+2x, equal to one another. Do that now.
Jenna
Luna
0
1
2
3
4
5
6
7
8
9
10
0
8
16
24
32
40
48
56
64
72
80
Time (weeks)
Number of miles
20+4x =
30+2x
Great job!
Let’s solve the equation for x, the number of weeks when Jenna and Luna run 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, 2x, while keeping the equation balanced?
Jenna’s miles = Luna's miles
20+4x = 30+2x
Multiply each side by 2x
Subtract 2x from each side
Add 2x to each side
Nice!
We can subtract 2x on both sides to get rid of the 2x on the right while keeping the equation balanced. Record subtracting 2x on both sides of the equation.
Jenna's miles
=
Luna's miles
20+4x
=
30+2x
−
2x
−
2x
Nice!
Subtract 2x on both sides and rewrite the equation to represent the remaining values.
Jenna's miles
=
Luna's miles
20+4x
=
30+2x
−
2x
−
2x
=
Great job!
We have all of the variables on the left. Now, keep going. What move can you make next to get rid of 20 on the left so that the x term, 2x, is alone on the left?
Jenna's miles
=
Luna's miles
20+4x
=
30+2x
−
2x
−
2x
20+2x
=
30
Add 2x on both sides.
Add 20 to both sides
Subtract 20 from both sides
You got it!
You can subtract 20 on both sides so that the x term remains alone on the left, and the equation is balanced. Record that in the equation.
Jenna's miles
=
Luna's miles
20+4x
=
30+2x
−
2x
−
2x
20 + 2x
=
30
−
20
−
20
Excellent!
Subtract 20 from both sides and rewrite the equation to represent the remaining values.
Jenna's miles
=
Luna's miles
20+4x
=
30+2x
−
2x
−
2x
20 + 2x
=
30
−
20
−
20
=
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.
Jenna's miles
=
Luna's miles
20+4x
=
30+2x
−
2x
−
2x
20 + 2x
=
30
−
20
−
20
2x
=
10
x
=
Excellent!
You just found that x = 5. 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 = 5? You can use either equation to find out. You can solve on paper, and then enter your answer on Zearn.
Miles Jenna runs per week: y= 20+4x
Miles Luna runs per week: y= 30+2x
y= 20+4x
y= 30+2x
When x= 5, y=
1 answer
To find the value of \( y \) when \( x = 5 \), we can use either equation. Let’s calculate it using both equations:
Using Jenna's equation: \[ y = 20 + 4x \] When \( x = 5 \): \[ y = 20 + 4(5) = 20 + 20 = 40 \]
Using Luna's equation: \[ y = 30 + 2x \] When \( x = 5 \): \[ y = 30 + 2(5) = 30 + 10 = 40 \]
In both cases, we find that when \( x = 5 \), \( y = 40 \).
So the corresponding y-coordinate is \( 40 \).
Thus the answer is: When \( x = 5, y = 40 \).
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