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Zorluk: OrtaSystems of Linear Equations

A chemist mixes a 10%10\% acid solution and a 30%30\% acid solution to create 200200 milliliters of a mixture that is 18%18\% acid. How many milliliters of the 30%30\% acid solution are in the mixture?

Cevap: 80 mL

Cevap

80
The correct answer is 8080 milliliters. By translating the problem into a system of linear equations, we let xx be the volume of the 10%10\% acid solution and yy be the volume of the 30%30\% acid solution. Since the total volume is 200200 milliliters, x+y=200x + y = 200. The amount of acid in the solutions must sum to the amount of acid in the final mixture, so 0.10x+0.30y=0.18(200)0.10x + 0.30y = 0.18(200), which simplifies to 0.10x+0.30y=360.10x + 0.30y = 36. Multiplying this equation by 1010 gives x+3y=360x + 3y = 360. Subtracting the first equation from this yields (x+3y)−(x+y)=360−200(x + 3y) - (x + y) = 360 - 200, or 2y=1602y = 160. Solving for yy gives 8080 milliliters of the 30%30\% solution.

Adım Adım Çözüm

1
Define variables and set up the system of equations.
Let xx be the number of milliliters of the 10%10\% acid solution and yy be the number of milliliters of the 30%30\% acid solution. The total volume equation is x+y=200x + y = 200. The total acid content equation is 0.10x+0.30y=0.18(200)0.10x + 0.30y = 0.18(200).
This sets up the system of linear equations representing the physical constraints of the mixture.
2
Simplify the acid content equation and prepare for elimination.
0.10x+0.30y=360.10x + 0.30y = 36. Multiplying the entire equation by 1010 yields x+3y=360x + 3y = 360.
Eliminating decimals simplifies the coefficients and makes it easier to solve the system using integer arithmetic.
3
Eliminate xx by subtracting the total volume equation from the simplified acid content equation.
(x+3y)−(x+y)=360−200  ⟹  2y=160  ⟹  y=80(x + 3y) - (x + y) = 360 - 200 \implies 2y = 160 \implies y = 80.
This isolates the variable yy, which directly represents the volume of the 30%30\% acid solution requested in the problem.

Anahtar Kavram

Solving systems of linear equations in context
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Systems of Linear Equations

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