1.1E: Exercises for Section 1.1

Exercise \(\PageIndex{1}\)

In each case verify that the following are solutions for all values of \(s\) and \(t\).

1. \[ \begin{array}[t]{rcl} x &=& 19t - 35 \\ y &=& 25 - 13t \\ z &=& t \end{array}\nonumber \] is a solution of \[ \begin{array}{rlrlrcr} 2x &+ & 3y & + & z & = & 5 \\ 5x &+ & 7y & - & 4z & = & 0 \end{array}\nonumber \]
2. \[ \begin{array}[t]{rcl} x_1 &=& 2s + 12t + 13\\ x_2 &=& s \\ x_3 &=& -s - 3t - 3\\ x_4 &=& t \end{array}\nonumber \] is a solution of \[ \begin{array}{rlrlrlrcr} 2x_1 & + & 5x_2 & + & 9x_3 & + & 3x_4 & = & -1 \\ x_1 & + & 2x_2 & + & 4x_3 & & & = & 1 \end{array}\nonumber \]

Answer

2. \(2(2s + 12t + 13) + 5s + 9(-s - 3t - 3) + 3t = -1\); \(2s + 12t + 13) + 2s + 4(-s - 3t - 3) = 1\)

Exercise \(\PageIndex{2}\)

Find all solutions to the following in parametric form in two ways.

1. \(3x + y = 2\)
2. \(2x + 3y = 1\)
3. \(3x - y + 2z = 5\)
4. \(x - 2y + 5z = 1\)

Answer

2. \(x = t\), \(y = \frac{1}{3}(1 - 2t)\) or \(x = \frac{1}{2}(1 - 3s)\), \(y = s\)
3. \(x = 1 + 2s - 5t\), \(y = s\), \(z = t\) or \(x = s\), \(y = t\), \(z = \frac{1}{5}(1 - s + 2t)\)

Exercise \(\PageIndex{3}\)

Regarding \(2x = 5\) as the equation \(2x + 0y = 5\) in two variables, find all solutions in parametric form.

Exercise \(\PageIndex{4}\)

Regarding \(4x - 2y = 3\) as the equation \(4x - 2y + 0z = 3\) in three variables, find all solutions in parametric form.

Answer

\(x = \frac{1}{4}(3 + 2s)\), \(y = s\), \(z = t\)

Exercise \(\PageIndex{5}\)

Find all solutions to the general system \(ax = b\) of one equation in one variable:

1. when \(a = 0\) and
2. when \(a \neq 0\).

Answer

1. No solution if \(b \neq 0\). If \(b = 0\), any \(x\) is a solution.
2. \(x = \frac{b}{a}\)

Exercise \(\PageIndex{6}\)

Show that a system consisting of exactly one linear equation can have no solution, one solution, or infinitely many solutions. Give examples

Exercise \(\PageIndex{7}\)

1. \( \begin{array}[t]{rlrcr} x & - & 3y & = & 5\\ 2x & + & y & = & 1 \end{array}\)
2. \( \begin{array}[t]{rlrcr} x & + & 2y & = & 0\\ & & y & = & 1 \end{array}\)
3. \( \begin{array}[t]{rlrlrcr} x & - & y & + & z & = & 2 \\ & & x & - & z & = & 1 \\ & & y & + &2x & = & 0 \end{array}\)
4. \( \begin{array}[t]{rlrcr} x & + & y & = & 1 \\ y & + & z & = & 0 \\ z & - & x & = & 2 \end{array}\)

Answer

2. \(\left[ \begin{array}{rr|r} 1 & 2 & 0 \\ 0 & 1 & 1 \end{array} \right]\)
3. \(\left[ \begin{array}{rrr|r} 1 & 1 & 0 & 1 \\ 0 & 1 & 1 & 0 \\ -1 & 0 & 1 & 2 \end{array} \right]\)

Exercise \(\PageIndex{8}\)

Write a system of linear equations that has each of the following augmented matrices.

1. \(\left[ \begin{array}{rrr|r} 1 & -1 & 6 & 0 \\ 0 & 1 & 0 & 3 \\ 2 & -1 & 0 & 1 \end{array} \right]\)
2. \(\left[ \begin{array}{rrr|r} 2 & -1 & 0 & -1 \\ -3 & 2 & 1 & 0 \\ 0 & 1 & 1 & 3 \end{array} \right]\)

Answer

2. \( \begin{array}{rlrlrcr} 2x & - & y & & & = & -1 \\ -3x & + & 2y & + & z & = & 0 \\ & & y & + & z & = & 3 \end{array}\) or \( \begin{array}{rlrlrcr} 2x_1 & - & x_2 & & & = & -1 \\ -3x_1 & + & 2x_2 & + & x_3 & = & 0 \\ & & x_2 & + & x_3 & = & 3 \end{array}\)

Exercise \(\PageIndex{9}\)

Find the solution of each of the following systems of linear equations using augmented matrices.

1. \( \begin{array}[t]{rlrcr} x & - & 3y & = & 1 \\ 2x & - & 7y & = & 3 \end{array}\)
2. \( \begin{array}[t]{rlrcr} x & + & 2y &=& 1 \\ 3x & + & 4y &=& -1 \end{array}\)
3. \( \begin{array}[t]{rlrcr} 2x & + & 3y &=& -1 \\ 3x & + & 4y &=& 2 \end{array}\)
4. \( \begin{array}[t]{rlrcr} 3x & + & 4y &=& 1 \\ 4x & + & 5y &=& -3 \end{array}\)

Answer

2. \(x = -3\), \(y = 2\)
3. \(x = -17\), \(y = 13\)

Exercise \(\PageIndex{10}\)

Find the solution of each of the following systems of linear equations using augmented matrices.

1. \( \begin{array}[t]{rlrlrcr} x & + & y & + & 2z & = & -1 \\ 2x & + & y & + & 3z & = & 0 \\ & - & 2y& + & z & = & 2 \end{array}\)
2. \( \begin{array}[t]{rlrlrcr} 2x & + & y & + & z & = & -1 \\ x & + & 2y & + & z & = & 0 \\ 3x & & & - & 2z & = & 5 \end{array}\)

Answer

b. \(x = \frac{1}{9}\), \(y = \frac{10}{9}\), \(z = -\frac{7}{3}\)

Exercise \(\PageIndex{11}\)

Find all solutions (if any) of the following systems of linear equations.

1. \( \begin{array}[t]{rcr} 3x - 2y & = & 5 \\ -12x + 8y & = & - 20 \end{array}\)
2. \( \begin{array}[t]{rcr} 3x - 2y & = & 5 \\ -12x + 8y & = & 16 \end{array}\)

Answer

b. No solution

Exercise \(\PageIndex{12}\)

Show that the system

\[ \left\{\begin{array}{r}
x+2 y-z=a \\
2 x+y+3 z=b \\
x-4 y+9 z=c
\end{array}\right. \nonumber \]

is inconsistent unless \(c = 2b - 3a\).

Exercise \(\PageIndex{13}\)

By examining the possible positions of lines in the plane, show that two equations in two variables can have zero, one, or infinitely many solutions.

Exercise \(\PageIndex{14}\)

In each case either show that the statement is true, or give an example² showing it is false.

1. If a linear system has \(n\) variables and \(m\) equations, then the augmented matrix has \(n\) rows.
2. A consistent linear system must have infinitely many solutions.
3. If a row operation is done to a consistent linear system, the resulting system must be consistent.
4. If a series of row operations on a linear system results in an inconsistent system, the original system is inconsistent.

Answer

2. F. \(x + y = 0\), \(x - y = 0\) has a unique solution.
3. T. Theorem 1.1.1.

Exercise \(\PageIndex{15}\)

Find a quadratic \(a + bx + cx^2\) such that the graph of \(y = a + bx + cx^2\) contains each of the points \((-1, 6)\), \((2, 0)\), and \((3, 2)\).

Exercise \(\PageIndex{16}\)

Solve the system \(\left \{ \begin{array}{rlrcr} 3x & + & 2y & = & 5 \\ 7x & + & 5y & = & 1 \end{array} \right.\) by changing variables \(\left \{ \begin{array}{rcrlr} x & = & 5x^\prime & - & 2y^\prime \\ y & = & -7x^\prime & + & 3y^\prime \end{array} \right.\) and solving the resulting equations for \(x^\prime\) and \(y^\prime\).

Answer

\(x^\prime = 5\), \(y^\prime = 1\), so \(x = 23\), \(y = -32\)

Exercise \(\PageIndex{17}\)

Find \(a\), \(b\), and \(c\) such that

\[\frac{x^2 - x + 3}{(x^2 + 2)(2x - 1)} = \frac{ax + b}{x^2 + 2} + \frac{c}{2x - 1} \nonumber \nonumber \]

[Hint: Multiply through by \((x^2 + 2)(2x - 1)\) and equate coefficients of powers of \(x\).]

Answer

\(a = -\frac{1}{9}\), \(b = -\frac{5}{9}\), \(c = \frac{11}{9}\)

Exercise \(\PageIndex{18}\)

A zookeeper wants to give an animal 42 mg of vitamin A and 65 mg of vitamin D per day. He has two supplements: the first contains 10% vitamin A and 25% vitamin D; the second contains 20% vitamin A and 25% vitamin D. How much of each supplement should he give the animal each day?

Exercise \(\PageIndex{19}\)

Workmen John and Joe earn a total of $24.60 when John works 2 hours and Joe works 3 hours. If John works 3 hours and Joe works 2 hours, they get $23.90. Find their hourly rates.

Answer

$4.50, $5.20

Exercise \(\PageIndex{20}\)

A biologist wants to create a diet from fish and meal containing 183 grams of protein and 93 grams of carbohydrate per day. If fish contains 70% protein and 10% carbohydrate, and meal contains 30% protein and 60% carbohydrate, how much of each food is required each day?