Belina buys 1 drink and one hotdog for $5. Georgia buys 2 drinks and 3 hotdogs for $13. How much does a drink cost? How much is each hotdog? Explain your reasoning.

Answer 1

You set variables for the price for one drink and price for one hotdog.

Let x be the price of one drink.

Let y be the price of one hotdog.

From the first sentence, 1 drink and 1 hotdog costs 5 dollars. 1 drink costs x dollars and 1 hotdog costs y dollars. Hence, we have our first equation.

x + y = 5

From the second sentence, 2 drinks and 3 hotdogs cost 13 dollars. Since 1 drink costs x dollars, 2 drinks costs 2x dollars. Also, since 1 hotdog costs y dollars, 3 hotdogs costs 3y dollars. Hence, we have our second equation.

2x + 3y = 13

We will solve for x by elimination. We do that by making the coefficient of y (the number in front of it) the same for both equations. This can easily be done by multiplying the first equation by 3. So we get

3x + 3y = 15

Now, we take the third equation minus the second equation:

3x + 3y = 15

-(2x + 3y) = 13

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1x + 0y = 2

So we get x = 2 meaning the price of one drink is 2 dollars. If y is also needed, you can always put our answer back in to any of the equations like the first one.

x + y = 5

2 + y = 5

y = 3

So the price of one hotdog is 3 dollars

Answer 2

Hello Gavin,

In these types of problems, learn to recognize when you have multiple unknowns. Anytime you have multiple unknowns you need to come up with those same number of equations to solve for those multiple unknowns.

In this case, we have two unknowns, the drink cost and the hotdog cost.

Because we have two unknowns, we have to come up with two equations to try to solve for those two unknowns.

We are going to identify the unknowns and assign them a variable

x is the cost of a drink

y is the cost of a hotdog

A general equation for Belina and Georgia is the number of drinks times the cost of a drink, plus the number of hotdogs times the cost of a hotdog equals the total amount that it cost them

For Belina the equation is

1($x) + 1($y) = $5

simplify it to x + y = 5

for Georgia, the equation is

2($x) + 3($y) = $13

simplify it to 2x + 3y = 13

Now we have two equations with two unknowns, we’re gonna solve for X which represents the drink cost, and y which represents the hotdog cost

x + y = 5

2x + 3y = 13

Now we will multiply through by -2 on the first equation and then add to the second equation so that the only variable left is y and then we can solve for y and then back substitute into one of the equations and solve for x

-2x - 2y = -10 and add to

2x + 3y = 13

and get

y = $3 for hotdog cost

Back substitute into original equation number one and solve for x

x + 3 = 5

so x = $2 for drink cost

Answer 3

Hi Gavin,

If x is the price of a drink and y is the price of a hotdog. As explained by other tutors, the fact that 1 drink and 1 hotdog cost $5 can be expressed as x + y = 5, and the fact that 2 drinks and 3 hotdogs costs $13 can be expressed as 2x+3y = 13. This is called a system of linear equations

There are 3 common ways to solve systems of linear equations.

Method 1: Graphing

This method involves graphing the two equations as lines to see where they intersect. While its not required, it's often easiest to graph equations when they are in slope-intercept form, so I'd recommend converting the equations to slope-intercept form (y = mx + b). You can do this by solving for y in both equations. By subtracting x from both sides in the first equation, we get y = -x+5. By subtracting 2x from both sides and dividing both sides by 3, we get y = (-2/3)x + (13/3). By graphing these two lines, we see they intersect at the point where x = 2 and y = 3.

Method 2: Substitution

In this method, we solve for one variable in one of the equations, and then substitute this expression into the other equation. We can start with either equation and either variable. For example, we can solve for y in x+y = 5. By subtracting x from both sides, we get y = 5 - x. Then we replace y with (5 - x) in the other equation:

2x + 3(5-x) = 13.

We can "multiply out" the first part to get 15 - 3x, so we have:

2x + 15-3x = 13.

Then we can combine the "x" terms to get:

15 - x = 13.

To solve for x, we can subtract 15 from both sides:

-x = -2

Finally, we multiply by -1:

x = 2.

So the price of a drink is $2. We can plug this back in to either equation to get the price of a hotdog, y. For example, when we replace x with 2 in the first equation, we get 2 + y = 5. If we subtract 2 from both sides, we get y = 3.

Method 3: Elimination

To use this method, we need to get the equations in a form so that they have the same term. For example, the second equation already has the term 2x, so we can multiply all terms in the first equation to get 2x + 2y = 10. Then we can subtract the equations like this:

2x + 2y = 10

-(2x + 3y = 13)

---------------------

-y = -3

If we multiply both sides by -1, we get y = 3. So the price of a hotdog is $3. Like in the substitution method, we can plug this back into either equation to solve for x. For example, if we use the first equation, we have x + 3 = 5. If we subtract 3 from both sides, we get x = 2.

Answer 4

Hi Gavin V

Since you can apply the method of solving two equations with two unknowns, you can set up both your in equations in Slope intercept form making them easy to graph and check

Slope Intercept form is y = mx + b

Solving for y will easily put both equations in this form

x = drinks

y = hotdogs

First equation is

x + y = 5

y = - x + 5

Second equation

2x + 3y = 13

3y = -2x + 13

y = (-1/3)(2x - 13)

To solve for x we can set the equations equal to one another, then to solve for y just plug the x value back into the first equation

-1/3(2x -13) = -x + 5

2x - 13 = -3(-x + 5)

2x - 13 = 3x - 15

15 - 13 = 3x - 2x

2 = x

We know from the first equation

y = -x + 5

y = 3

Drinks are $2 each and hotdogs are $3 each

You can graph your equations at Desmos.com, on a graphing calculator, or by hand and check the point of intersection to confirm our answer.

Answer 5

X = drink

Y = Hot dog

X + Y = $5

2x + 3y = $13

Solve for y first (solve for hotdog)

Start with "X + Y = $5" and subtract Y from both sides, you're left with the equation below:

X = $5 - Y

Now that you know what "X" also known as "drink" equals, you can plug it into the second equation to solve for Y:

2($5-Y) + 3y = $13 ---> $10 - 2y + 3y = $13 ---> $10 + y = $13 ---> y = $3

Now that we know that hotdog is $3, we can plug it into the first equation to figure out the price of the drink

X + $3 = $5 ---> X = $2

To conclude, the drink is $2 and the hot dog is $3.