Multi-Step Equations: Apply
Show What You Know
Solve the following equation showing all your steps.
2(x+4.5)=−3
Solve the following equation showing all your steps.
6.2=−3.1(y+3.1)
José bought three sweaters and two shirts, and he paid a total of $111. The sweaters cost $7 more than the shirts. What would the total cost be if he decided to buy another sweater? Explain the process you used to get the answer. Interpret your solution for the context of the problem.
Hal and Sal are planning their wedding. The Gold Ring Catering Company has given a quote for food for 95 people at the reception of $1,955, which includes $22 for each adult and $13 for each child. Their friends Katarina and Melanie recently had a change of plans. They can now come to the wedding and will bring their two children. How many adults and children originally planned to attend the wedding? How much more will Hal and Sal have to pay if they add two adults and two children to their wedding bill? Explain all the steps in the process you used to answer the question. Interpret your solution for the context of the problem.
1) Solve the equation 2(x+4.5) = -3:
First, distribute 2 to the terms inside the parentheses:
2x + 9 = -3
Then, isolate the variable by subtracting 9 from both sides:
2x = -12
Finally, divide both sides by 2 to solve for x:
x = -6
So the solution to the equation is x = -6.
2) Solve the equation 6.2 = -3.1(y+3.1):
First, distribute -3.1 to the terms inside the parentheses:
6.2 = -3.1y - 9.61
Then, combine like terms by adding 9.61 to both sides:
15.81 = -3.1y
Next, isolate the variable by dividing both sides by -3.1:
y = -15.81/-3.1
Lastly, simplify the division to find the value of y:
y ≈ 5.109
So the solution to the equation is y ≈ 5.109.
3) Let's first set up some equations based on the given information:
Let S be the cost of one sweater and C be the cost of one shirt.
According to the problem, we know that:
3S + 2C = 111 (total cost equation)
S = C + 7 (sweater cost equation)
To find the total cost if José decides to buy another sweater, we need to increase the number of sweaters bought by 1.
So the new total cost equation would be:
4S + 2C = ?
To solve for ?, we substitute the value of S from the sweater cost equation into the total cost equation:
4(C + 7) + 2C = ?
4C + 28 + 2C = ?
6C + 28 = ?
6C = ? - 28
C = (? - 28)/6
We need the value of ?, which represents the new total cost. However, the problem does not provide enough information to determine this value. Therefore, we cannot find the exact total cost if José decides to buy another sweater.
4) Let's set up equations based on the given information:
Let A be the number of adults and C be the number of children originally planned to attend the wedding.
According to the problem, we know that:
A + C = 95 (total number of people equation)
22A + 13C = 1955 (total cost equation)
To find the number of adults and children originally planned to attend the wedding, we need to solve the system of equations.
First, let's solve for A by isolating A in the first equation:
A = 95 - C
Next, substitute this value of A into the second equation:
22(95 - C) + 13C = 1955
Distribute 22 to both terms inside the parentheses:
2090 - 22C + 13C = 1955
Combine like terms:
-9C = 1955 - 2090
-9C = -135
Divide both sides by -9 to solve for C:
C = -135/-9
C = 15
Now that we know the number of children, we can substitute this value into the first equation to solve for A:
A + 15 = 95
A = 95 - 15
A = 80
Therefore, the number of adults originally planned to attend the wedding is 80 and the number of children originally planned to attend is 15.
To calculate how much more Hal and Sal will have to pay if they add two adults and two children to their wedding bill, we need to find the new total cost.
New total number of people = 95 + 2 adults + 2 children
New total number of people = 95 + 2(80) + 2(15)
New total number of people = 95 + 160 + 30
New total number of people = 285
New total cost = 22(adults) + 13(children)
New total cost = 22(80) + 13(15)
New total cost = 1760 + 195
New total cost = 1955
Therefore, Hal and Sal will have to pay an additional $1955 if they add two adults and two children to their wedding bill.