Nonlinear Systems

Main idea

Once a curve like a circle or parabola is involved, substitution is the tool: it collapses two equations into one you can solve. Curves can cross more than once, so expect several solutions, or none.

\begin{aligned} y &= x + 1 \\ x^2 + y^2 &= 25 \end{aligned} \implies x^2 + {\htmlData{sym=0}{(x + 1)^2}} = 25

What the symbols mean

$(x + 1)^2$
Substituted y
Replacing y with x + 1 leaves one equation in x.

Explanation

A nonlinear system is simply a system where at least one equation isn't a straight line, like a circle or a parabola. ("Nonlinear" just means "not linear.") The question hasn't changed: which points lie on every graph at once? The formula up top shows the whole trick in one line: $y = x + 1$ (a line) tells you what $y$ is, so you swap it into $x^2 + y^2 = 25$ (a circle) and get $x^2 + (x + 1)^2 = 25$ , one equation with only $x$ in it. It's the substitution you already know.

Why substitution, not elimination? Elimination needed matching terms to cancel, and with an $x^2$ in one equation and a plain $x$ in the other there's usually nothing to match. Substitution doesn't care: it always trades two equations for one, and then you're solving something you've seen before, often a quadratic.

Method:

Solve the simpler equation for one variable.
Substitute into the other and solve the resulting single-variable equation.
Find the partner value for each solution, using the simpler equation.
Check each pair in both equations.

Counting solutions. Sketch first, so you know what to expect. A line can miss a circle, touch it once, or cut through it twice, so a line and a circle give 0, 1 or 2 solutions. Two conics (circles, parabolas, ellipses, hyperbolas) can cross in up to 4 points.

A common mistake. When the single-variable equation gives two $x$ values, it's tempting to stop there, or to find just one $y$ . Each $x$ is half of a separate point, so each needs its own partner $y$ , and it's safest to get it from the simpler equation. Plugging into $x^2 + y^2 = 25$ instead would hand you $\pm$ values, and only one sign actually lies on the line.

See it

This is how cell towers locate a phone. It's 5 km from tower A and r km from tower B, which sits 6 km away, so the phone is on both circles. Depending on r, they cross twice, touch once, or miss. Subtracting the two circle equations cancels the squares and leaves the dashed line x = (61 − r²)/12, so the nonlinear system becomes a linear one. A third tower settles which crossing is right.

The line y = x + 1 cuts right through the circle x² + y² = 25, crossing it at (3, 4) and (−4, −3). Two crossings means two solutions.

Worked examples

Example 1

Solve $y = x + 1$ , $x^2 + y^2 = 25$ .

1
The line already gives $y$ , so substitute it into the circle: $x^2 + (x + 1)^2 = 25$ , so $2x^2 + 2x - 24 = 0$ and $x^2 + x - 12 = 0$ .

Practice

Type your answer and check it. Solve them all to complete the lesson.

Problem 1+5 XP

Solve $y = x + 1$ , $x^2 + y^2 = 25$ . Enter the solution with positive $x$ .

Problem 2+5 XP

Enter the other solution.

Problem 3+5 XP

Find the x-coordinates where $y = x^2$ meets $y = 2x + 3$ .

Problem 4+5 XP

How many solutions does $x^2 + y^2 = 4$ , $y = 5$ have?

Problem 5+5 XP

Solve $xy = 6$ , $x + y = 5$ . Enter the solution with $x < y$ .

Problem 6+5 XP

A phone is 5 km from tower A at $(0, 0)$ , 5 km from tower B at $(6, 0)$ , and 5 km from tower C at $(0, 8)$ . Where is it?

Problem 7+5 XP

Solve $y = x^2 - 2x + 2$ , $y = 2x - 2$ . Enter $x$ , $y$ .

Challenge problem+25 XP

The hardest one here. Get it, and you've got this lesson.

A rectangle has a diagonal of 5 cm and an area of 12 cm². Find its side lengths, longer first.

Where it's used

Finding where a projectile's path crosses a slope, where two orbits intersect, or the equilibrium of nonlinear supply and demand curves.

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