Systems with Lines and Curves, SAT math lesson

A thrown ball follows a curved path while a drone climbs in a straight line. When are they at the same height at the same moment? Questions like this ask where a line meets a curve. Two lines meet at most once, but a line and a parabola can meet twice, once or not at all, and the algebra has to find every one of those points.

See also Curve Intersections and Which Curve Is Higher in Foundations for Calculus and Nonlinear Systems and Curve Intersections in Precalculus.

Key terms

Nonlinear system

Definition A system of equations in $x$ and $y$ is a set of equations considered together. It is nonlinear when at least one of its equations is not linear, that is, cannot be written as $Ax + By = C$ with constants $A$ , $B$ and $C$ where $A$ and $B$ are not both zero. Typical SAT examples pair a line with a parabola $y = ax^2 + bx + c$ or with a reciprocal relation $xy = k$ .

In plain words A nonlinear system asks for the points that lie on a line and a curve at the same time, or on two curves.

Example The system

\begin{aligned} y &= x^2, \\[0.4em] y &= x + 2 \end{aligned}

is nonlinear because the first equation contains $x^2$ . The point $(2, 4)$ satisfies both, since $2^2 = 4$ and $2 + 2 = 4$ .

Simultaneous solution

Definition A solution of a system in $x$ and $y$ is an ordered pair $(x, y)$ of real numbers that makes every equation of the system true at once. The solution set is the set of all such ordered pairs. When the variables stand for quantities, only pairs inside the stated domain count.

In plain words One equation is not enough. The pair has to pass every equation, and the answer is a pair, not just an $x$ -value.

Example A ball’s height is $h = -5t^2 + 10t + 15$ meters for $0 \le t \le 3$ seconds, and a drone’s height is $h = 5t + 5$ meters. Setting the heights equal gives

\begin{aligned} -5t^2 + 10t + 15 &= 5t + 5, \\[0.4em] t^2 - t - 2 &= 0, \\[0.4em] (t - 2)(t + 1) &= 0. \end{aligned}

The algebra gives $t = 2$ and $t = -1$ , but $t = -1$ is outside $0 \le t \le 3$ . At $t = 2$ both heights are $15$ , since $-20 + 20 + 15 = 15$ and $10 + 5 = 15$ . The only solution is $(t, h) = (2, 15)$ , so they are level after $2$ seconds at a height of $15$ meters.

Substitution

Definition To solve a system by substitution, solve one equation for one variable, replace that variable in the other equation, and solve the resulting equation in one variable. For each solution, put it back into an original equation to recover the other coordinate, then check the pair in both original equations. If both equations have the form $y = f(x)$ and $y = g(x)$ , substitution gives the single equation $f(x) = g(x)$ .

In plain words Two expressions for the same $y$ must be equal where the graphs meet. Set them equal, solve for $x$ , then go back for $y$ .

Example Solve the system $y = x^2 - 4x + 5$ and $y = x + 1$ . Setting the two expressions equal gives

\begin{aligned} x^2 - 4x + 5 &= x + 1, \\[0.4em] x^2 - 5x + 4 &= 0, \\[0.4em] (x - 1)(x - 4) &= 0. \end{aligned}

So $x = 1$ or $x = 4$ , and the line gives $y = 2$ or $y = 5$ . Checking in the parabola, $1 - 4 + 5 = 2$ and $16 - 16 + 5 = 5$ . The solutions are $(1, 2)$ and $(4, 5)$ .

Intersection

Definition A point of intersection of two graphs is a point that lies on both. The intersection points of the graphs of the equations in a system are exactly the real solutions of the system. For a line $y = mx + n$ and a parabola $y = ax^2 + bx + c$ with $a \ne 0$ , substitution gives the quadratic equation

ax^2 + (b - m)x + (c - n) = 0.

Each real root gives exactly one intersection point, so the graphs meet in two points, one point or no point according to whether the discriminant of this quadratic is positive, zero or negative.

In plain words Count the real solutions of the substituted equation and you have counted the meeting points. A negative discriminant means the line misses the parabola entirely.

Example For $y = x^2 + 3$ and $y = 2x$ , substitution gives $x^2 - 2x + 3 = 0$ , with discriminant $(-2)^2 - 4 \cdot 3 = -8$ . There is no real solution, and the line passes below the parabola without touching it. A graph is a useful check but not a proof. In the Desmos graphing calculator you can type y=x^2 and y=x+1 and click an intersection point to see its coordinates, but they are rounded decimals, close to $(1.618, 2.618)$ for one of the two points. The exact values come from solving $x^2 = x + 1$ , which gives

x = \frac{1 \pm \sqrt{5}}{2}.

A graph can also hide a meeting point outside the visible window or blur two points that are very close together.

Tangent intersection

Definition For a parabola $y = ax^2 + bx + c$ with $a \ne 0$ , a line is tangent to the parabola when it is not vertical and meets the parabola in exactly one point. This one-point test does not work for every curve. It is not the general meaning of tangent, since for some curves a line can meet the curve once without touching it, or touch it at one point and cross it somewhere else. For a nonvertical line and a parabola, tangency happens exactly when the substituted quadratic has discriminant zero, so its one real root is a double root. The system then has exactly one distinct solution.

In plain words The line just touches the curve. The quadratic factors as a perfect square, and the touching point counts once.

Example For $y = x^2$ and $y = 2x - 1$ , substitution gives

x^2 - 2x + 1 = (x - 1)^2 = 0,

so $x = 1$ and $y = 1$ . The system has the single solution $(1, 1)$ , and the line touches the parabola there. A vertical line such as $x = 1$ also meets the parabola only once, but it crosses rather than touches, which is why the definition excludes it.

Line and reciprocal intersection

Definition For a nonzero constant $k$ , the relation $xy = k$ forces both coordinates to be nonzero, since $x = 0$ or $y = 0$ would make the product $0$ . So on its graph you may write $y = \frac{k}{x}$ with $x \ne 0$ . To intersect it with a line, substitute the line into $xy = k$ , solve, and recall that no solution can have a zero coordinate.

In plain words The curve $xy = k$ never touches either axis. Substituting a slanted line into it gives a quadratic, so there can be two, one or no meeting points. A horizontal line $y = c$ with $c \ne 0$ meets it exactly once, at $x = \frac{k}{c}$ , by crossing it rather than touching it, and the axes never meet it at all.

Example Solve $xy = 6$ and $y = x + 1$ . Substitution gives

\begin{aligned} x(x + 1) &= 6, \\[0.4em] x^2 + x - 6 &= 0, \\[0.4em] (x + 3)(x - 2) &= 0. \end{aligned}

So $x = 2$ with $y = 3$ , or $x = -3$ with $y = -2$ . Both products are $6$ , and both points lie on the line. The solutions are $(2, 3)$ and $(-3, -2)$ .

Check yourself

Which ordered pair is a solution of the system $y = x^2 - 2x$ and $y = x + 4$ ? (A) $(4, 0)$ , (B) $(-1, 3)$ , (C) $(1, 5)$ , (D) $(-1, -3)$

Show the answer

The correct choice is (B). Setting $x^2 - 2x = x + 4$ gives $x^2 - 3x - 4 = 0$ , so $x = 4$ or $x = -1$ . The solutions are $(4, 8)$ and $(-1, 3)$ . Choice (A) has a correct $x$ -value but takes $y$ from $x - 4$ instead of $x + 4$ , and (D) uses the right $x$ with a sign error in $y$ . Choice (C) lies on the line but not on the parabola, where $1 - 2 = -1$ .

In the system $y = x^2 + 4$ and $y = bx$ , the constant $b$ is positive and the system has exactly one solution. What is the value of $b$ ?

Show the answer

The answer is $4$ . Substitution gives $x^2 - bx + 4 = 0$ , which has exactly one real solution when its discriminant is zero, $b^2 - 16 = 0$ . So $b = 4$ or $b = -4$ , and the condition $b > 0$ leaves $b = 4$ . The line $y = 4x$ touches the parabola at $(2, 8)$ .

How many solutions does the system $xy = -4$ and $y = x + 4$ have? (A) $0$ , (B) $1$ , (C) $2$ , (D) $4$

Show the answer

The correct choice is (B). Substitution gives $x(x + 4) = -4$ , so

x^2 + 4x + 4 = (x + 2)^2 = 0.

The only root is $x = -2$ , which gives $y = 2$ , and indeed $(-2)(2) = -4$ . The double root means the line touches the curve at $(-2, 2)$ . Choice (C) assumes every quadratic gives two points, (A) would need a negative discriminant, and (D) confuses the count with the constant $4$ .

Systems with Lines and Curves

Theory, part 67

Practice, part 68

Key terms

Nonlinear system

Simultaneous solution

Substitution

Intersection

Tangent intersection

Line and reciprocal intersection

Check yourself

How this lesson connects

Builds on

Used later in