The relationships between x, the number of weeks, and y, the number of miles Steven and Irene run each week, are graphed on the coordinate plane below. What is the point of intersection?
Steven
Irene
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 (
,
).
Great job!
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 Steven and Irene are shown below. Since the number of miles Steven runs will be equal to the number of miles Irene 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.
Steven
Irene
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
Excellent!
Let’s solve the equation for x, the number of weeks when Steven and Irene 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?
Steven’s miles = Irene's miles
20+4x = 30+2x
Multiply each side by 2x
Subtract 2x from each side
Add 2x to each side
You got it!
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.
Steven's miles
=
Irene's miles
20+4x
=
30+2x
−
2x
−
2x
Excellent!
Subtract 2x on both sides and rewrite the equation to represent the remaining values.
Steven's miles
=
Irene's miles
20+4x
=
30+2x
−
2x
−
2x
=
Nice!
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?
Steven's miles
=
Irene's miles
20+4x
=
30+2x
−
2x
−
2x
20+2x
=
30
Add 2x on both sides.
Subtract 20 from both sides
Add 20 to both sides
1 answer
To isolate the \( x \) term on the left side of the equation, you can subtract 20 from both sides of the equation. This will help you simplify the equation further.
Let's write that step down:
Starting with:
\[ 20 + 4x = 30 + 2x \]
Subtracting \( 20 \) from both sides:
\[ 20 + 4x - 20 = 30 + 2x - 20 \]
This simplifies to:
\[ 4x = 10 + 2x \]
Great! Now, what's the next step to isolate \( x \)?
