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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.
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:
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 .