System of Equations
I need help with the following problem:
A youth group with 26 members is going skiing. Each of the five chaperones will drive a van or a sedan. The vans can seat 7 people and the sedans can seat five people. How many of each type of vehicle could transport all 31 people to the ski area in one trip?
I used the following equation:
7x + 5y = 31. (x= van, y=sedan)
7x=31-5y.
I divdied by 7 on both sides.
x=31-5y/7.
This is where I got stuck. What am I doing wrong?
You have only one equation. You need another.
x + y = 5 (vans + sedans = 5)
Next, you make an error in your last statement.
7x=31-5y so
x = (31-5y)/7
However, I would do it an easier way unless your teacher told you to solve by substitution. I would solve by elimination.
x+y=5
7x+5y=31
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Multiply top equation by 7
7x+7y=35
7x+5y=31
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Now subtract equation 2 from equation 1 to obtain
2y=4 with y = 2, then solve for x.
Well, it seems like you're almost there! The equation you set up is correct: 7x + 5y = 31, where x represents the number of vans and y represents the number of sedans.
To solve for the values of x and y, you need another equation. In this case, the equation represents the fact that there are a total of 26 people in the youth group:
x + y = 26.
Now you have a system of two equations:
7x + 5y = 31
x + y = 26
Solving this system should give you the values of x and y, which represent the number of vans and sedans, respectively.
Remember, math problems can be tricky, but with a little humor, you'll get through it like a clown in a circus tent!
You made a small mistake when dividing both sides of the equation by 7. Let me correct it for you.
Starting from 7x = 31 - 5y, let's divide both sides by 7:
7x/7 = (31 - 5y)/7
Simplifying the left side gives:
x = (31 - 5y)/7
So far, you are correct.
Now, remember that x represents the number of vans and y represents the number of sedans. Since both x and y must be integers (as fractional vehicles are not possible), we can use trial and error to find the values of x and y that satisfy the equation.
We know that the total number of vehicles must be less than or equal to 5, as there are only 5 chaperones available. Also, the total number of people transported must be 31.
Let's try different values for x and find the corresponding y:
- If x = 1, then substituting into the equation gives: 1 = (31 - 5y)/7. Solving this equation, we get y = 2.
So, if there is 1 van and 2 sedans, the total number of people transported would be 7 + 2*5 = 17. This falls short of the required 31 people.
- If x = 2, then substituting into the equation gives: 2 = (31 - 5y)/7. Solving this equation, we get y = 3.
So, if there are 2 vans and 3 sedans, the total number of people transported would be 2*7 + 3*5 = 29, which is still less than 31.
- If x = 3, then substituting into the equation gives: 3 = (31 - 5y)/7. Solving this equation, we get y = 4.
So, if there are 3 vans and 4 sedans, the total number of people transported would be 3*7 + 4*5 = 41. However, this exceeds the required 31 people.
From the above calculations, we see that no combination of vans and sedans can transport all 31 people in one trip.
Therefore, the answer to the problem is there is no solution for the given conditions.
You are on the right track with setting up the equation, but you made a mistake when you divided both sides of the equation by 7. Let me explain how to correctly set up and solve this system of equations.
Let's let x represent the number of vans and y represent the number of sedans needed to transport all 31 people.
From the given information, we know that each van can seat 7 people, so the number of people transported by vans will be 7x. Similarly, each sedan can seat 5 people, so the number of people transported by sedans will be 5y.
Since the total number of people is 31, we can write the first equation as:
7x + 5y = 31
Now, we need to consider the constraints. We know that there are 5 chaperones, each driving either a van or a sedan. So the total number of vehicles is equal to the number of chaperones, which is 5:
x + y = 5
Now we have a system of equations:
Equation 1: 7x + 5y = 31
Equation 2: x + y = 5
To solve this system, we can use the method of substitution. We can solve Equation 2 for either x or y and substitute that value into Equation 1. Let's solve Equation 2 for x:
x = 5 - y
Substituting this value of x into Equation 1, we get:
7(5 - y) + 5y = 31
35 - 7y + 5y = 31
-2y = 31 - 35
-2y = -4
Dividing both sides of the equation by -2, we get:
y = -4 / -2
y = 2
Now, substitute the value of y back into Equation 2 to find the value of x:
x + 2 = 5
x = 5 - 2
x = 3
So, the solution to the system of equations is x = 3 and y = 2. Therefore, you would need 3 vans and 2 sedans to transport all 31 people to the ski area in one trip.