Question

solve the system. if there are infinitely many solutions, write each answer as a function of t like ( x = x(t), y = y(t) ) where ( z = t ). if there are no solutions, type dne in each answer box.
( \begin{cases} 2x - 3y - 9z = - 14 \\ x + 3z = 2 \\ - 3x + y - 4z = 0 end{cases} )
( x=)
( y=)
( z=)
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question 18
solve the system. if there are infinitely many solutions, write each answer as a function of t like ( x = x(t), y = y(t) ) where ( z = t ). if there are no solutions, type dne in each answer box.
( \begin{cases} x + y + z = 4 \\ y - 3z = 5 \\ 2x + y + 5z = 4 end{cases} )
( x=)
( y=)
( z=)
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question 19
a chemist has three different acid solutions. the first acid solution contains ( 25% ) acid, the second contains ( 35% ) third contains ( 80% ). he wants to use all three solutions to obtain a mixture of 60 liters containing ( 55% ) acid, times as much of the ( 80% ) solution as the ( 35% ) solution. how many liters of each solution should be used?

Explanation:

Step 1: Solve for \(x\) from the second equation

From \(x + 3z=2\), we can express \(x\) as \(x = 2-3z\).

Step 2: Substitute \(x = 2 - 3z\) into the first and the third equations

• Substitute into \(2x-3y - 9z=-14\):

$$
LATEXBLOCK0
$$

• Substitute into \(-3x + y-4z = 0\):

$$
LATEXBLOCK1
$$

Since \(z=t\) (given in the problem - statement for the format of the answer), then:

• \(x=2-3t\) (from \(x = 2-3z\) with \(z = t\))
• \(y=6-5t\) (from \(y=6 - 5z\) with \(z = t\))
• \(z=t\)

Answer:

\(x = 2-3t\), \(y=6 - 5t\), \(z=t\)