A chemistry laboratory prepares a 100\text{ mL} mixture using three solutions: Solution X … — GRE General Test

A chemistry laboratory prepares a $100\text{ mL}$ mixture using three solutions: Solution $X$ ( $10\%$ acid by volume), Solution $Y$ ( $30\%$ acid by volume), and Solution $Z$ ( $50\%$ acid by volume). The resulting mixture is $37\%$ acid by volume. If the volume of Solution $Y$ used is $5\text{ mL}$ more than twice the volume of Solution $X$ , what is the volume, in $\text{mL}$ , of Solution $Z$ used in the mixture?

A
$15$
B
$35$
C
$40$
$50$ Cevap
E
$55$

Cevap

50\text{ mL}

Setting up the system of equations gives

x + y + z = 100

x + 3y + 5z = 370

, and

y = 2x + 5

. Substituting

y = 2x + 5

into the first two equations yields

3x + z = 95

and

7x + 5z = 355

. Solving for

x

gives

x = 15

, which leads to

z = 95 - 3(15) = 50

. Therefore,

50\text{ mL}

of Solution Z was used.

Adım Adım Çözüm

Define variables and set up the system of linear equations

Let

x

y

, and

z

be the volumes in

\text{mL}

of Solutions

X

Y

, and

Z

, respectively.
1) Total volume:

x + y + z = 100

2) Total acid volume:

0.10x + 0.30y + 0.50z = 0.37(100) \implies x + 3y + 5z = 370

3) Relationship between

Y

and

X

y = 2x + 5

Translate the word problem statements into mathematical equations.

Substitute

y = 2x + 5

into equations (1) and (2) to reduce to a two-variable system

From equation (1):

x + (2x + 5) + z = 100 \implies 3x + z = 95 \implies z = 95 - 3x

From equation (2):

x + 3(2x + 5) + 5z = 370 \implies 7x + 15 + 5z = 370 \implies 7x + 5z = 355

Eliminating

y

simplifies the system to two equations in

x

and

z

Substitute

z = 95 - 3x

into

7x + 5z = 355

and solve for

x

7x + 5(95 - 3x) = 355 \implies 7x + 475 - 15x = 355 \implies -8x = -120 \implies x = 15

Solves for the unknown volume of Solution X.

Calculate

z

using

z = 95 - 3x

z = 95 - 3(15) = 95 - 45 = 50

Finds the requested volume of Solution Z.

Anahtar Kavram

Setting up and solving 3x3 systems of linear equations using substitution or elimination

Tahmini Süre:2m 0s

Soru

Cevap

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