Question

100 POINTS!!!
Could you answer the following questions using elimination please

Answer 1

Explanation:

7. To find the number of pumpkins and squash sold at the vegetable stand, we can set up a system of equations based on the given information. Let's say the number of pumpkins sold is represented by P, and the number of squashes sold is represented by S. From the problem, we know that squashes were sold 6 more than pumpkins: S = P + 6. We also know that the total sales amounted to $98. The cost of each pumpkin is $5, and the cost of each squash is $3. So, we can write an equation for the total sales as: 5P + 3S = 98. Now we can solve this system of equations to find the values of P and S. Substituting the value of S from the first equation into the second equation, we get: 5P + 3(P + 6) = 98. Simplifying the equation, we have: 5P + 3P + 18 = 98. Combining like terms, we get: 8P + 18 = 98. Subtracting 18 from both sides, we have: 8P = 80. Dividing both sides by 8, we find: P = 10. So, the number of pumpkins sold is 10. Substituting this value back into the first equation, we get: S = 10 + 6, which gives us S = 16. Therefore, the vegetable stand sold 10 pumpkins and 16 squashes. 8. To find the number of hours of overtime Ramiro worked on the weekend, we can set up a system of equations based on the given information. Let's say the number of hours Ramiro worked during the week is represented by W, and the number of hours worked on the weekend is represented by E. From the problem, we know that Ramiro earned $20 per hour during the week and $30 per hour for overtime on the weekend. We also know that Ramiro worked 5 times as many hours during the week as he did on the weekend: W = 5E. We also know that Ramiro earned a total of $650. So, we can write an equation for his earnings as: 20W + 30E = 650. Now we can solve this system of equations to find the values of W and E. Substituting the value of W from the second equation into the first equation, we get: 20(5E) + 30E = 650. Simplifying the equation, we have: 100E + 30E = 650. Combining like terms, we get: 130E = 650. Dividing both sides by 130, we find: E = 5. So, Ramiro worked 5 hours of overtime on the weekend. Substituting this value back into the second equation, we get: W = 5(5), which gives us W = 25. Therefore, Ramiro worked 25 hours during the week and 5 hours of overtime on the weekend.

Answer 2

7. They sold 10 pumpkins and 16 squashes.

8. Ramiro worked 5 hours of overtime.

Explanation:

Question #7
To determine the number of pumpkins and squashes sold at the vegetable stand, we'll establish a system of equations based on the given information and then use the elimination method to solve it.

Let's denote:

• p = number of pumpkins sold

• s = number of squashes sold

From the information:

1. One day they sold 6 more squash than pumpkins.

• s = p + 6

2. Their sales totaled $98.

• 5p + 3s = 98

Thus, we have the following system of equations:

`\left\{\begin{array}{ccc}s=p+6& (1)\\5p+3s=98 &(2)\end{array}\right`

To eliminate 's' using the elimination method, multiply the first equation by 3 to make the coefficient of 's' equal in both equations:

`\Longrightarrow \left\{\begin{array}{ccc}3[s=p+6]& (1)\\5p+3s=98 &(2)\end{array}\right\\\\\\\Longrightarrow \left\{\begin{array}{ccc}3s=3p+18& (1)\\5p+3s=98 &(2)\end{array}\right\\\\\\\\\Longrightarrow \left\{\begin{array}{ccc}3p-3s=-18& (1)\\5p+3s=98 &(2)\end{array}\right`

Now, add equation (1) and (2) together:

`\Longrightarrow [3p-3s=-18]+[5p+3s=98]\\\\\\\\\Longrightarrow (3p+5p)+(-3s+3s)=-18+98\\\\\\\Longrightarrow 8p=80`

Solve for 'p':

`\Longrightarrow p=(80)/(8) \\\\\\\\\therefore \boxed{p=10}`

Now take p = 10 and substitute it into equation (1) to find 's':

`\text{Recall, } s=p+6\\ \\ \\ \\ \Longrightarrow s=10+6\\\\\\\\\therefore \boxed{s=16}`

Therefore, they sold 10 pumpkins and 16 squashes.

`\hrulefill`

Question #8

To determine how many hours of overtime Ramiro worked on the weekend, we'll set up a system of equations based on his earnings and then apply the elimination method to solve.

Let's denote:

• w = number of hours Ramiro worked during the week

• o = number of hours Ramiro worked overtime on the weekend

From the information:

1. He worked 5 times as many hours during the week as he did on the weekend.

• w = 5o

2. One week Ramiro earned a total of $650.

• 20w + 30o = 650

Thus, we have the following system of equations:

`\left\{\begin{array}{ccc}w=5o& (1)\\20w+30o=650 &(2)\end{array}\right`

To eliminate 'w' using the elimination method, multiply the first equation by 20 to make the coefficient of 'w' equal in both equations:

`\Longrightarrow \left\{\begin{array}{ccc}20[w=5o]& (1)\\20w+30o=650 &(2)\end{array}\right\\\\\\\\\Longrightarrow \left\{\begin{array}{ccc}20w=100o& (1)\\20w+30o=650 &(2)\end{array}\right\\\\\\\\\Longrightarrow \left\{\begin{array}{ccc}-20w+100o=0& (1)\\20w+30o=650 &(2)\end{array}\right`

Now, add equation (1) and (2) together:

`\Longrightarrow [-20w+100o=0]+[20w+30o=650]\\\\\\\\\Longrightarrow (-20w+20w)+(100o+30o)=0+650\\\\\\\\\Longrightarrow 130o=650`

Solve for 'o':

`\Longrightarrow o =(650)/(130)\\\\\\\\\therefore \boxed{o=5}`

Therefore, Ramiro worked 5 hours of overtime on the weekend.