# Homework on Linear Systems of Equations Solutions

**Purpose:** To provide you with more practice on
reduced row-echelon form and on the parametric representations

of solutions to systems of linear equations .

1. Find an equation for the three -dimensional linear object which has three
direction vectors

and and
which passes through the point P(7, 5, 3, 1).

**Answer:** The equation is given by .
Thus, the

parametric form is w = s + 7, x = −2s − 7t + 5, y = 3s + 2t + 3, and z = 5s + 5t
+ 1.

2. Find the parametric form of the equation for a line in 4-D which passes
through the points P(1, 2, 3, 4)

and Q(−3, 5, 4, 1).

**Answer:** We can use as the direction
vector and Q as the position to

obtain , or w = −4t−4, x = 3t+3, y = t+1, and
z = −3t−3.

3. Complete the following two problems :

(a) Determine three distinct ways to balance the following chemical equation:
.

In particular, find the balanced equation in
which two (2) moles of are produced

for every five (5) moles of consumed.

**Answer:**

Consider . Solving the resulting system of

equations results in the following matrix:

Therefore, there are two free variables , D = s and E = t,
and the rest are determined by

A = 2s−t, B = −s+t, and C = 2s+t.We can get several distinct solutions. For
example,

A = 1, B = 1, C = 7, D = 2, E = 3 and A = 2, B = 1, C = 10, D = 3, E = 4 and

A = 1, B = 2, C = 11, D = 3, E = 5 and A = 3, B = 1, C = 13, D = 4, E = 5. In

the second, the ratio between water produced and hydrogen consumed is 2 to 5.

(b) Using systems of equations , balance the following chemical equation:

**Answer: **

4. For each of the following augmented matrices:

(a) Tell whether it is in row- reduced form .

(b) If not, state which part of the definition it violates and put it in
row- reduced form .

(c) Identify the dimension of the solution space from the reduced matrix.

(d) Write the parametric form and the vector parametric form of the solution to
the system of equations .

**Answer:**

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