Systems with a Parameter

Three Possible Outcomes

Each equation $ax + by = c$ is a line. Two lines in a plane can only do three things: cross once, stay parallel, or lie on top of each other.

The quickest clue is the slope. Lines with different slopes always cross exactly once. Lines with the same slope are either parallel (different y-intercepts) or the same line (same y-intercept).

The Determinant Test

Comparing slopes means dividing by a coefficient that might be zero for some parameter value. There is a cleaner way that never divides.

Slide $m$ below and watch the second line. At $m = -2$ it lands exactly on top of the first one.

Special Values: Parallel or Same Line?

When you solve $D = 0$, each solution is a special value of the parameter. At that value the lines have equal slopes, and you must decide: same line or parallel?

Sometimes the parameter appears only in a constant, as in $2x - y = a$ and $-4x + 2y = 8$. Then $D = (2)(2) - (-4)(-1) = 0$ for every $a$, so the parameter in the constants decides everything. Make the equations multiples of each other: the second one is $-2 \times (2x - y = -4)$, so you need $a = -4$ for the lines to coincide. For any other $a$ the lines are parallel.

Writing the infinitely many solutions

When a system has infinitely many solutions, the exercise may ask you to describe them all. Both equations say the same thing, so keep just one of them. Solve it for one variable, then call the other variable a free parameter such as $\lambda$ (or $t$), which can be any real number.

Choose as the free variable the one that keeps the other expression free of fractions. If the discussion shows that no value of the parameter gives infinitely many solutions, say so: the answer to "describe the infinite solutions" is then "this never happens."

A Full Discussion, Step by Step

Put the pieces together into one routine you can use on any system with one parameter.

Two Parameters

Sometimes one parameter sits in the coefficients and another in the constants. The determinant decides whether there is a unique solution (it depends only on the coefficients). The second parameter only matters at the special values.

The picture below shows every pair $(k, n)$. Only two points give infinitely many solutions.

Lines and Real-World Situations

Each equation of the system is a line, so the three outcomes have a geometric meaning: intersecting (one solution), parallel (none), or coincident (infinitely many). A parameter might stand for wind speed, a concentration, or a resistance. The math is identical; you just state the result in words at the end.

Let's close the hook. The drone paths are $2x + vy = 6$ and $vx + 2y = 6$. The determinant is $D = 4 - v^2 = (2 - v)(2 + v)$, which is zero at $v = 2$ and $v = -2$.

• For $v \ne \pm 2$ the paths cross at exactly one point.

• For $v = 2$ both equations become $2x + 2y = 6$, so the drones fly the same path.

• For $v = -2$ the equations become $x - y = 3$ and $x - y = -3$, so the paths are parallel and never meet.

If the situation only allows $v \ge 0$ (say, wind speed measured as a positive number), then $v = -2$ is not possible. The context can remove cases, but you still discuss the system first and then interpret.

When the System Looks Different

Three situations need a small preparation step before the determinant test works.

For the second situation, recall $\sin^2 \alpha + \cos^2 \alpha = 1$. For the system $(\cos t)x + (\sin t)y = 1$ and $-(\sin t)x + (\cos t)y = 2$, the determinant is $\cos^2 t + \sin^2 t = 1$. It is never zero, so there is exactly one solution for every angle $t$.

For the third, expand $D$ completely before deciding. For example, $D = k^2 - (k + 3)(k - 3) = k^2 - (k^2 - 9) = 9$. The parameter cancels, so $D \ne 0$ always and the system always has one solution.

Non-Linear Systems with a Parameter

When one equation is a parabola or a circle, there is no determinant. The question becomes: how many intersection points are there? Substitute the line into the other equation and you get a quadratic. Then the discriminant $\Delta = b^2 - 4ac$ decides.

The same plan works for a circle. For $x^2 + y^2 = 5$ and $y = 2x + c$, substitution gives $5x^2 + 4cx + c^2 - 5 = 0$ with $\Delta = 100 - 4c^2$. So there are two points when $|c| < 5$, one point when $|c| = 5$, and none when $|c| > 5$.

Back to the start: the drone paths $2x + vy = 6$ and $vx + 2y = 6$ have $D = 4 - v^2$. They cross at one point unless $v = 2$ (same path) or $v = -2$ (parallel paths that never meet).

Summary