00:01
Hi, my name is avi and we're here to solve problem 67 from chapter 10 systems of equations and inequalities of section 1, systems of lanier equations and 2 variables.
00:14
So let's quickly read the questionnaire here.
00:17
Nutrition a researcher performs an experiment to test the hypothesis that involves the nutrients niacin and retinol.
00:24
She feeds one group of laboratory rats a daily diet of precisely 32 units of niacetosis.
00:31
And 22 ,000 units of retinal.
00:34
Let's quickly note that down.
00:36
So 32 units of niacin and 22 ,000 units of ret retinel.
01:08
She uses two types of commercial pellet foods.
01:11
Food a contains 0 .12 unit of niacin and 100 units of retinal per gram.
01:18
Food b contains 0 .2 unit of and 50 units of retinal per gram.
01:24
How many grams of each food does she feed this group of rats each day? so, note this down.
01:33
Food a contains 0 .12 units of niacin and 100 units of retinol.
02:09
Food b contains 0 .2 units of niacin and 50 units of retinol.
02:41
And this is all in one gram of each of the foods.
02:51
And the question is, how many grams of each food, basically, does she feed the rats to get 32 units of niacin? and 22 ,000 units of retinol.
03:11
So to solve this, first let's give each of them variables.
03:17
So let's give food a niacin.
03:20
So let's give food a x.
03:22
So x will count the number of grams of food a used.
03:27
So number of grams food a.
03:40
And then we'll have y be the number of grams for food b.
03:58
So looking at this, we can already set up a system of equations based on the information given to us.
04:05
So since we know that there is 0 .12 units of niacin in one gram, food a, we can go ahead and set that equal to 0 .12x, because that's the number of niacin in 1 gram, food a, grams, x will be the number of grams of food a.
04:26
So we can add that to point 12, sorry, point 20 y, which is the number of grams of food b.
04:38
And we can set that all equal to 32, 32 units of niacin.
04:50
Basically what this means is this point 12 comes from the food a up here.
04:57
There are 0 .12 units of niacin in one gram.
05:02
This is the x value here, which is the multiplier of how many grams of food a there are going to be in 32 units of niacin that this researcher gives to the rats.
05:14
And the same thing for food b, there are 0 .20 units of niacin and y is the number of grams.
05:23
Conversely, we can set up another set of equations using retinal.
05:26
So since we know that there is 100 units of retinal in food a in one gram, we can go ahead and add that to the 50 units of retinol in one gram in one gram of food b.
05:50
And we know that if we add these together, we should reach 22 ,000 units of retinal in total.
06:10
Now, what we're trying to do here is solve for x and y to find out how many grams of each food is the research you're using.
06:19
To do this, we can do the substitution method to set these two equations so we can find x and y.
06:27
To do that, i'm going to go ahead and select this equation and plug it in to here.
06:35
To do that, we have to solve for one of the variables first.
06:38
So i'll pick y.
06:39
So let me quickly create a new page.
06:46
Unwrite the equations one more time.
06:49
So 0 .12x plus 0 .20y should equal 32 units of niacin.
07:13
And the other equation was 100 units of retinol of food a, plus 50 units of retinol from food b should equal 22 ,000 units of retinol in total.
07:46
So now i said we were going to do the substitution method, and i said i was going to select this equation to plug into here.
07:54
So first i'm going to have to solve for a variable.
07:57
So 100x plus 50y equals 22.
08:09
I'm going to go ahead and subtract 100x on both sides to isolate one term...