Problem 1

we will balance the following chemical equation by finding the least positive integer coefficients

for each molecule.

MnS+As₂ Cr ₂ O35 +H₂SO4 → HMnO4+As H3 +CrS3 O₁2+H₂O.

To begin, we introduce coefficients on each molecule:

x₁MnS+x₂ As₂ Cr ₁00 35+X3 H₂SO4 → X₁ HMnO4+x5 As H₂+x CrS3 0₁2 + x, H₂O,

to balance the equation, we need to find numerical values for these coefficients.

We know that the number of atoms of each type must be the same on both sides of the chemical

equation. Therefore, we obtain an equation for each type of atom: \begin{align} \text{As:} &&

2x_2 &=x_5||text{Cr:} && 10x_2 &=x_6||text{H} && 2x_3&=x_4 + 3x_5+2x_7||text{Mn:} &&

x_1&=x_4 | \text{0:} && 35x_2 +4x_3&= 4x_4 + 12x_6 +x_7| \text{S:} && x_1+x_3&= 3x_6|

end{align}

We can rearrange the wquations to obtain the following homogeneous system: \begin{array}

{rrrrrrrl} & 2x_2 & & & -x_5 & & & =0\ & 10x_2 & & & & -x_6 & & =0 \ & & 2x_3 & -x_4 & -3x_5 &

& -2x_7 & =0\x_1 & & & -x_4 & & & & =01 & 35x_2 & +4x_3 & -4x_4 & & -12x_6 & -x_7 & =0\x_1

& & +x_3 & & & -3x_6 & & =0 \end{array}

Part (a)

Form the coefficient matrix of this system./nPart (b)

Form the augmented matrix and use the command rref[] to get it into the correct form.

Part (c)

Give the solution set of this system.

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Answer:

Is this a consistent system?

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Answer:

How many solutions does it have?

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Answer:

Which variables are basic and which variables are free?

(Input your answer by double clicking and typing in the cell below.)/nPart (d)

Find a nontrivial solution that has all positive, integral values.

Part (f)

Check that your solution does indeed satistfy the equation. (i.e., compute the product of the

coefficient matrix with your solution and check that the answer is the zero vector).

Part (g)

Give the final balanced equation.

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Answer: