The relationships between x, the number of weeks, and y, the number of miles Cameron and Samara run each week, are graphed on the coordinate plane below. What is the point of intersection?
Cameron
Samara
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 Cameron and Samara are shown below. Since the number of miles Cameron runs will be equal to the number of miles Samara 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.
Cameron
Samara
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
Nice!
Let’s solve the equation for x, the number of weeks when Cameron and Samara 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?
Cameron’s miles = Samara's miles
20+4x = 30+2x
Add 2x to each side
Subtract 2x from each side
Multiply each side by 2x
1 answer
To get rid of the x term on the right side of the equation (which is \(2x\)), you can subtract \(2x\) from each side.
So the equation:
\[ 20 + 4x = 30 + 2x \]
becomes:
\[ 20 + 4x - 2x = 30 \]
This simplifies to:
\[ 20 + 2x = 30 \]
Now, you can solve for \(x\). What would you like to do next?
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