# Yuri & Kaoru's Weight Loss Journey

## The Situation...

Yuri and Kaoru, both age 12, have been obese most of their lives. One day their doctor tells them that their lives will be in danger, if they don't drastically lose weight over the course of one year. Yuri with a medium frame, currently weighs 180 lbs and has a height of 5'1". Kaoru also with a medium frame, currently weighs 212 lbs and has a height of 5'5". According, to their doctor Yuri has to weigh from 120-110 lbs and Kaoru has to weigh from 140-120 lbs in order to be healthy, so he puts both of them on a specific exercise and diet plan and records their weight per month.

Yuri loses 3 pounds each month.

Kaoru loses 7 pounds each month.

When will Yuri and Kaoru's weight be the same?

## Variables

Dependent Variable- Weight

Independent Variable- Months Passed

Starting Weight: 180 lbs. (Yuri) and 212 lbs. (Kaoru)

Weight Lost Per Month: 3 lbs. (Yuri) and 7 lbs. (Kaoru)

y= -3x+180

y= -7x+212

## Desmos Graphic Solution

Graphical Solution:

Yuri: y=-3x+180

1. Plot the y-intercept which is (0,180).

2. From that point, use the slope (-3/1 or -3) to graph the line

Kaoru: y=-7x+212

1. Plot the y-intercept which is (0,212).

2. From that point, use the slope (-7/1 or -7) to graph the line

Now you can look at the graph a find the point of intersection/solution, which is (8,156).

You can also find the solution with the process of substitution-

Yuri: y=-3x+180

Kaoru: y=-7x+212

If y=y, then that means...-3x+180 = -7x+212

-3x+180 = -7x+212

-3x+7x = 212-180

4x = 32

x = 8

Now we can substitute x into one of the equations to find y

y=-3(8)+180

y=-24+180

y=156

So now you know that x=8 and y=156, therefore the point of intersection is (8,156).

You can also find the solution with the process of elimination-

Yuri: y=-3x+180

Kaoru: y=-7x+212

You know that these two equation both have y in common, so we're going to eliminate by making one of the equation negative and subtracting in from the other. This way the positive and negative y, will cancel out.

y= -3x +180

-(y= -7x +212)

0= 4x - 32

32 = 4x

32/4 = x

x=8

Now that we know what x is, we can substitute x in one of the equation to find y.

y= -3(8) +180

y= -24 + 180

y= 156

Therefore the point of intersection/solution is (8,156).

## Explanation

The point of intersection/solution is important because it shows the doctor, that 8 months from the beginning of their weight loss journey, Yuri and Kaoru will weigh the same (156 lbs.). The solution will show that before 8 months had passed Yuri weighed less than Kaoru and after 8 months she weighs more than Kaoru. This way the doctor now knows that he has to put Yuri on a stricter exercise and meal plan, in order for her to be safe and healthy. This shows the significance of the solution, as without it the doctor wouldn't have known to put Yuri on a stricter exercise and meal plan.

## All in all...

Yuri and Kaoru will both weigh 156 pounds in 8 months, after the beginning of their weight loss.