Model Contexts with Two Variable Expressions Q2

Question

Kevin went to the grocery store and bought bottles of soda and bottles of juice. Each bottle of soda has 30 grams of sugar and each bottle of juice has 40 grams of sugar. Kevin purchased a total of 12 bottles of juice and soda which collectively contain 450 grams of sugar. Algebraically determine the number of bottles of soda purchased and the number of bottles of juice purchased.

Answer 1

Hi Katie, this is a classic example of modeling real world problems with a system of linear equations. Let us begin by representing the number of bottles of juice by J and the number of bottles of soda with S. Now our goal is to write two linear equations in terms of J and S from the context of the problem.

Since Keven bought 12 total drinks, we have the equation S + J = 12. We also know how much sugar is in each drink, as well as the total amount of sugar in all the drinks purchased. The equation 30S + 40J = 450 represents this scenario in terms of S and J. Now we have two equations with two unknowns (S and J) that make up the following linear system of equations:

S+J =12

30S+40J =450

We can solve this system by using either elimination or substitution (I will use substitution here):

S+J = 12

S=12- J

Now we can substitute the expression 12- J in for S in the second equation, and then solve for J:

30(12- J) +40J = 450

360- 30J +40J = 450

10J = 90

J = 9

Finally, we can substitute 9 in for J in either of the two original equations to solve for S:

S + 9 = 12

S = 3

So Keven bought three bottles of soda and nine bottles of juice. Hope this helps. Let me know if you have any questions.

Answer 2

This problem can easily be solved by using a systems of equations and solve that systems either by substitution , elimination or graphically.

I am going to set up the system by calling x= bottle of sodas purchased and y= bottles of juice purchased.

X. +. Y = 12. ........(1). equation #1 I am going to use the method of "Elimination "to solve the

30X + 40Y=450. ........(2) equation #2 systems of equations.

________________

I am going to multiply equation #1 by -30 (coefficient of the x) and equation #2 by 1 (coefficient of X)

(-30). X + Y = 12

(1) 30X +40Y=450

_________________

-30X -30Y + -360

30X +40Y= 450

______________

subtracting equation (1) and equation (2) we got:

10Y = 90

Y= 90/10 =9. bottles of juice.

Now we need to find the bottles of soda purchased .

X +. Y= 12

X + 9 =12

subtracting 9 in both sides

we got

X= 3. bottles of Soda. Solution: ( x, y)=. (3, 9).

Remember you can always check your work by substitute x and y in either equation #1 or equation #2

3+9 =12

30(3) + 40(9)=450

Answer 3

Let the number of soda bottle bought be x.

Let the number of juice bottle bought be y.

Equation 1: Total number of bottle is x + y and it is given that Kevin bought 12 bottles. Therefore, our first equation is

x + y = 12

Equation 2: Total sugar in soda bottles 30x grams and total sugar in juice bottles is 40y grams and it is given that the total amount of sugar in all bottles is 450 grams. Therefore, our second equation is

30x + 40y = 450

Dividing by 10, we have

3x + 4y = 45

We solve by the method of substitution. Solving equation 1 for y, we get

y = 12 - x

Now plugging this value of y in equation 2, we get

3x + 4(12-x) = 45

3x + 48 - 4x = 45

- x = - 3

x = 3

Plugging x = 3 in y = 12 - x, we get y = 9.

Therefore, Kevin bought 3 soda bottles and 9 juice bottles.

Answer 4

We know that Kevin is buying a TOTAL of 12 bottles of drinks. So for example, if he bought 4 bottles of juice, that means that he would buy 8 bottles of soda because 12-4 gives you 8.

Now, to set this up algebraically, you use the same kind of logic. So, suppose that Kevin buys X bottles of soda. That means that he buy 12-X bottles of juice. From the problem, we know that sugar for soda is 30 grams per bottle and the sugar for juice is 40 grams per bottle and the total sugar for the 12 bottles is 450 grams. This gives us the equation:

X(30) + (12-X)(40) = 450

because we have X bottles of soda with each having 30 grams of sugar and 12-X bottles of juice with each having 40 grams of sugar for a total of 450 grams of sugar for all 12 bottles.

Simplifying the equation give us:

30X + 480 -40X= 450, which gives us that X = 3. This means that Kevin buys 3 bottles of Soda and then 12-3= 9 bottles of juice.