The relationships between x, the number of weeks, and y, the number of miles Tyler and Emma run each week, are graphed on the coordinate plane below. What is the point of intersection?
Tyler
Emma
0
1
2
3
4
5
6
7
8
9
10
0
3
6
9
12
15
18
21
24
27
30
Time (weeks)
Number of miles
The point of intersection is (
,
).
Good work!
Now that you have found the point of intersection, (4, 18), let’s check that work algebraically. The equations for the relationship between weeks and miles for Tyler and Emma are shown below. Since the number of miles Tyler runs will be equal to the number of miles Emma runs at the solution to this system, you can set the expression for the number of miles they each run, 10+2x and 30−3x, equal to one another. Do that now.
Tyler
Emma
0
1
2
3
4
5
6
7
8
9
10
0
3
6
9
12
15
18
21
24
27
30
Time (weeks)
Number of miles
10+2x
=
30−3x
Good work!
Let’s solve the equation for x, the number of weeks when Tyler and Emma 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, 3x, while keeping the equation balanced?
Tyler’s miles = Emma's miles
10+2x = 30−3x
Add 3x to each side
Multiply each side by 3x
Subtract 3x from each side
1 answer
To get rid of the x term on the right side of the equation \(30 - 3x\), you can add \(3x\) to each side of the equation. This will help isolate the \(x\) terms on one side. Here’s how it looks step by step:
Starting with the equation:
\[ 10 + 2x = 30 - 3x \]
Add \(3x\) to both sides:
\[ 10 + 2x + 3x = 30 - 3x + 3x \]
This simplifies to:
\[ 10 + 5x = 30 \]
Now you can solve for \(x\) from here.
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