Answers to Tutorial on Systems of Linear Equations

Detailed solutions to the questions in the Systems of Linear Equations tutorial are presented below.

Question 1: Finding a Specific Point of Intersection

Set the constants \(a_1\), \(a_2\), \(b_1\), \(b_2\), \(c_1\), and \(c_2\) so that the two lines have a specific point of intersection \((x, y)\). Verify that this ordered pair is the solution to the system.

Solution 1

As an example, let \(a_1 = 1\), \(b_1 = 2\), \(c_1 = 2\), \(a_2 = 1\), \(b_2 = 1\), and \(c_2 = 0\). The resulting system is: \[ \begin{cases} x + 2y = 2 \\ x + y = 0 \end{cases} \]

For these coefficients, the graphical point of intersection is \((-2, 2)\). Substituting \(x = -2\) and \(y = 2\) satisfies both equations.

Solving by substitution: From the second equation, \(y = -x\). Substituting into the first: \[ x + 2(-x) = 2 \implies -x = 2 \implies x = -2 \] Then \(y = -(-2) = 2\). The analytical solution \((-2, 2)\) matches the graphical solution.

Question 2: Parallel Lines and the Determinant

Set \(a_1\), \(a_2\), \(b_1\), and \(b_2\) such that \(a_1 b_2 - a_2 b_1 = 0\). How are the lines positioned? How many solutions exist? What does this imply about their slopes?

Solution 2

Example: Let \(a_1 = 1\), \(b_1 = 1\), \(c_1 = 0\), \(a_2 = 3\), \(b_2 = 3\), and \(c_2 = 6\). Then: \[ a_1 b_2 - a_2 b_1 = (1)(3) - (3)(1) = 0 \] The system is: \[ \begin{cases} x + y = 0 \\ 3x + 3y = 6 \end{cases} \]

The two lines are parallel. The system has no solution (inconsistent).

If \(a_1 b_2 - a_2 b_1 = 0\), then \(\frac{a_1}{b_1} = \frac{a_2}{b_2}\) (assuming \(b_1, b_2 \neq 0\)). This means the lines have equal slopes and are therefore parallel (or coincident if also \(c_1/c_2\) equals the same ratio).

Question 3: Coincident Lines and Infinite Solutions

Set all six constants so that \(a_1/a_2 = b_1/b_2 = c_1/c_2\). How many solutions does the system have?

Solution 3

Example: Let \(a_1 = 2\), \(b_1 = 2\), \(c_1 = 2\), \(a_2 = 1\), \(b_2 = 1\), and \(c_2 = 1\). Then: \[ \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} = 2 \] The system is: \[ \begin{cases} x + y = 1 \\ 2x + 2y = 2 \end{cases} \]

The second equation is simply the first multiplied by 2. They are equivalent equations and represent the same line. The system has infinitely many solutions (all points on the line).