Exercise 40

When plotting the following system of equations:

\[ \begin{cases} 0.5 x + y = 2 \\[.5em] 0.4 x + 2y = 2 \end{cases} \]

we obtain two straight lines that are:

  1. perpendicular to each other

  2. orthogonal to each other

  3. parallel and coincident

  4. parallel and distinct

  5. secant

Try to solve it before checking the answer.

  1. secant

Recall that the two lines that graphically represent the system are:

\[ \begin{cases} ax + by = c \\[.5em] a’x + b’ y = c’ \end{cases} \]

Let \(L_2\) be the line representing the equation \( 0.4 x + 2y = 2 \).

To plot it, let’s find the points where the line intersects the coordinate axes:

\[ \begin{aligned} x = 0 &\Rightarrow 0.4 (0) + 2y = 2 \\[.5em] &\Rightarrow y = 1 \\[.5em] &\Rightarrow L_2 \, \text{ passes through } \, (0, \, 1) \\[1em] y = 0 &\Rightarrow 0.4 x + 2(0) = 2 \\[.5em] &\Rightarrow x = \frac{2}{0.4} \\[.5em] &\Rightarrow x = 5 \\[.5em] &\Rightarrow L_2 \, \text{ passes through } \, (5, \, 0) \end{aligned} \]

Similarly we proceed with \(L_1\), which will represent the equation \( 0.5x + y = 2 \). We obtain that \(L_1\) passes through the points \( (0, \, 2\) and \( (4, \, 0\).

The plots tell us that these lines are not perpendicular or orthogonal, nor are they distinct or coincident parallels. They are simply two secant lines.

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