Examkin

Soru

Zorluk: OrtaSystems of Linear Equations
A landscape designer purchased a total of 3535 plants, consisting of boxwood shrubs and fern plants, for a total of \510$. Each boxwood shrub cost \18 ,andeachfernplantcost$, and each fern plant cost \$ 12 .Ifthedesignerpurchased. If the designer purchased b boxwoodshrubsand boxwood shrubs and f fernplants,whatisthevalueof fern plants, what is the value of f - b$?
  1. A
    15
  2. 5Cevap
  3. C
    27
  4. D
    29

Cevap

The correct answer is 5, representing the difference between the 20 fern plants and 15 boxwood shrubs.
To find the value of f−bf - b, we can set up a system of linear equations based on the given information. The total number of plants purchased is 3535, which gives the equation b+f=35b + f = 35. The total cost of the plants is \510$, with boxwood shrubs costing \18 eachandfernplantscosting$ each and fern plants costing \$ 12 each,givingtheequation each, giving the equation 18b + 12f = 510 .Solvingthefirstequationfor. Solving the first equation for f gives gives f = 35 - b .Substitutingthisintothesecondequationyields. Substituting this into the second equation yields 18b + 12(35 - b) = 510 .Distributingandsimplifyinggives. Distributing and simplifying gives 18b + 420 - 12b = 510 ,whichsimplifiesto, which simplifies to 6b + 420 = 510 .Subtracting. Subtracting 420 frombothsidesgives from both sides gives 6b = 90 ,so, so b = 15 .Substituting. Substituting b = 15 backinto back into b + f = 35 gives gives 15 + f = 35 ,so, so f = 20 .Thevalueof. The value of f - b is is 20 - 15 = 5$.

Adım Adım Çözüm

1
Define variables and write the system of linear equations representing the problem.
Let bb be the number of boxwood shrubs and ff be the number of fern plants. The system of equations is:
b+f=3518b+12f=510\begin{aligned} b + f &= 35 \\ 18b + 12f &= 510 \end{aligned}
We need to translate the word problem into a mathematical system of two linear equations with two variables.
2
Express one variable in terms of the other from the first equation and substitute it into the second equation.
From b+f=35b + f = 35, we get f=35−bf = 35 - b.
Substituting this into the second equation gives:
18b+12(35−b)=51018b + 12(35 - b) = 510
Substitution is a standard method to reduce a system of two equations to a single equation in one variable.
3
Solve the resulting single-variable equation for bb.
18b+420−12b=51018b + 420 - 12b = 510
6b+420=5106b + 420 = 510
6b=906b = 90
b=15b = 15
By distributing the coefficient and combining like terms, we isolate and solve for the variable b.
4
Substitute the value of bb back into the first equation to solve for ff, and then calculate f−bf - b.
15+f=35⇒f=2015 + f = 35 \Rightarrow f = 20
f−b=20−15=5f - b = 20 - 15 = 5
Finding the individual values of both variables allows us to calculate the required difference.

Anahtar Kavram

Solving systems of linear equations in two variables using substitution or elimination, and evaluating a linear combination of the solution.

Alternatif Yöntem

Instead of substitution, the system can be solved using elimination. Multiply the first equation, b+f=35b + f = 35, by 1212 to get 12b+12f=42012b + 12f = 420. Subtract this from the second equation, 18b+12f=51018b + 12f = 510, to eliminate ff: (18b−12b)+(12f−12f)=510−420(18b - 12b) + (12f - 12f) = 510 - 420, which simplifies to 6b=906b = 90, giving b=15b = 15. Then, find f=20f = 20 and compute f−b=5f - b = 5.
Tahmini Süre:1m 30s
Bu soruyu puanla

Konular

Systems of Linear Equations

Bu konudan daha fazla soru