Solving Equations by Graphing Both Sides

Quick answer

An equation f(x) = g(x) asks where two outputs agree. Graph y = f(x) and y = g(x) on the same axes and the answer is visible: the x-coordinates of the crossing points. That works for any pair of functions, including pairs no algebra at this level can solve, such as 2 to the x equals x plus 3. A table then pins each crossing down to as many decimal places as needed.

What you'll learn

Explain why intersection points give the solutions of f(x) = g(x)
Approximate solutions from a graph and a table of values
Narrow a solution by successive approximation

One equation, two graphs

The equation

2^x = x + 3

has no algebraic method at this level. The unknown sits in an exponent on one side and in a sum on the other, and no rearrangement isolates it.

But the equation asks a plain question: for which $x$ do these two expressions give the same number? Graph each side as its own function and the question becomes visual.

y = 2^x \qquad\text{and}\qquad y = x + 3

y = 2^x
y = x + 3

Graphing each side of 2^x = x + 3

The graphs cross twice, so the equation has two solutions: about $-2.86$ and about $2.44$ .

Why the crossing points are the solutions

A graph is the set of every point that satisfies its equation. So:

$(a, b)$ is on the graph of $y = f(x)$ exactly when $f(a) = b$
$(a, b)$ is on the graph of $y = g(x)$ exactly when $g(a) = b$

A point on both graphs therefore has $f(a) = b$ and $g(a) = b$ . Two things equal to the same number are equal to each other:

f(a) = g(a)

So the $x$ -coordinate of every crossing is a solution.

The argument also runs the other way. If $f(a) = g(a)$ , call the shared value $b$ . Then $(a, b)$ is on the first graph and on the second, so it is a crossing point. The solutions and the crossings are the same list — none missing and none extra.

Nothing in that argument used the kind of function. It works for lines, parabolas, exponentials, absolute values, reciprocals and logarithms alike.

Narrowing with a table

A graph gives a solution to about a tenth. A table does better.

Subtract one side from the other. A solution is where $2^x - (x + 3) = 0$ , so look for the difference changing sign.

$x$	$2^x$	$x + 3$	difference
$2.3$	$4.925$	$5.3$	$-0.375$
$2.4$	$5.278$	$5.4$	$-0.122$
$2.5$	$5.657$	$5.5$	$+0.157$
$2.6$	$6.063$	$5.6$	$+0.463$

The difference goes from negative to positive between $2.4$ and $2.5$ . At $2.4$ the line is still above the curve; at $2.5$ the curve has passed it. The crossing is in between.

Repeat with a step ten times smaller.

$x$	$2^x$	$x + 3$	difference
$2.43$	$5.3889$	$5.43$	$-0.0411$
$2.44$	$5.4264$	$5.44$	$-0.0136$
$2.45$	$5.4642$	$5.45$	$+0.0142$
$2.46$	$5.5022$	$5.46$	$+0.0422$

Now the sign changes between $2.44$ and $2.45$ . Each round buys one more decimal place, and the rounds can continue as long as needed. This is successive approximation.

Other kinds of functions

The method needs only two graphs. Absolute value works the same way.

|x - 1| = 3 - \tfrac{x}{2}

y = |x − 1|
y = 3 − x/2

Graphing each side of |x − 1| = 3 − x/2

The line meets each arm of the V once, so there are two solutions. Reading them and checking:

|-4 - 1| = 5 = 3 + 2 \qquad \left|\tfrac{8}{3} - 1\right| = \tfrac{5}{3} = 3 - \tfrac{4}{3}

A graph also reports how many solutions to expect before any algebra starts. Here the answer is two, so an algebraic solution that finds only one has missed a case.

Worked examples

Common mistakes

Practice problems

The graphs of $y = f(x)$ and $y = g(x)$ cross at $(2, 7)$ and $(-1, 4)$ . Solve $f(x) = g(x)$ .

Answer

$x = 2$ or $x = -1$

Full solution

The solutions are the $x$ -coordinates of the crossings. The $y$ -coordinates, $7$ and $4$ , are the shared values.
The graphs of $y = x^2$ and $y = 2x + 3$ cross at $(-1, 1)$ and $(3, 9)$ . Solve $x^2 = 2x + 3$ .

Answer

$x = -1$ or $x = 3$

Full solution

Read the $x$ -coordinates, then check: $(-1)^2 = 1 = 2(-1) + 3$ ✓ and $3^2 = 9 = 2(3) + 3$ ✓
How many real solutions does $x^2 = -4$ have?

Answer

None.

Full solution

$y = x^2$ never goes below $0$ , so it never meets the line $y = -4$ .
Between which two consecutive whole numbers is the solution of $2^x = 20$ ?

Answer

Between $4$ and $5$ .

Full solution

$2^4 = 16 < 20$ and $2^5 = 32 > 20$ , so the curve passes $y = 20$ between $x = 4$ and $x = 5$ .
For $2^x = x + 3$ , the difference $2^x - (x + 3)$ is $-0.122$ at $x = 2.4$ and $+0.157$ at $x = 2.5$ . What does that tell you?

Answer

A solution lies between $2.4$ and $2.5$ .

Full solution

The difference changes sign, so the two sides trade places between those inputs. They must be equal somewhere in between.
Solve $\tfrac{1}{x} = x$ and explain how the graph shows there are exactly two solutions.

Answer

$x = 1$ or $x = -1$ .

Full solution

The line $y = x$ passes through each branch of $y = \tfrac{1}{x}$ once, at $(1, 1)$ and $(-1, -1)$ .

Neither graph meets the other anywhere else, so those are the only solutions.
Check that $x = -4$ and $x = \tfrac{8}{3}$ both solve $|x - 1| = 3 - \tfrac{x}{2}$ .

Answer

Both check.

Full solution

At $x = -4$ : $|-5| = 5$ and $3 + 2 = 5$ ✓

At $x = \tfrac{8}{3}$ : $\left|\tfrac{5}{3}\right| = \tfrac{5}{3}$ and $3 - \tfrac{4}{3} = \tfrac{5}{3}$ ✓
Find the solution of $3^x = 5$ to one decimal place.

Hint

Try $x = 1.4$ and $x = 1.5$ .

Answer

About $1.5$ .

Full solution

$3^{1.4} \approx 4.656$ and $3^{1.5} \approx 5.196$ , so the solution is between $1.4$ and $1.5$ .

To decide which way it rounds, test the midpoint: $3^{1.45} \approx 4.918$ , still below $5$ . So the solution is above $1.45$ and rounds to $1.5$ .
Explain why the argument that crossings are solutions works for every kind of function.

Answer

It uses only what a graph means — every point satisfies its equation — and never the formula of either function.

Full solution

The argument says: a point on both graphs has $f(a) = b$ and $g(a) = b$ , so $f(a) = g(a)$ . And if $f(a) = g(a)$ , the point $(a, f(a))$ lies on both graphs.

Neither step asks whether $f$ is linear, exponential or anything else. So the method applies to any pair of functions that can be graphed.
Reading the crossing $(3, 9)$ of $y = x^2$ and $y = 2x + 3$ , Andre reports that the solution of $x^2 = 2x + 3$ is $9$ . Find his errors.

Hint

Which coordinate is the input? And is there only one crossing?

Answer

He reported the $y$ -coordinate, and he missed a crossing. The solutions are $x = 3$ and $x = -1$ .

Full solution

At $(3, 9)$ the input is $3$ and the shared output is $9$ . The equation asks for the input, so this crossing gives $x = 3$ .

Substituting $9$ shows the mistake: $9^2 = 81$ but $2(9) + 3 = 21$ .

The parabola also crosses the line at $(-1, 1)$ , since $(-1)^2 = 1 = 2(-1) + 3$ . So the full answer is $x = 3$ or $x = -1$ .

Frequently asked questions

Why do intersection points solve f(x) = g(x)?

A point on both graphs has one x and one y. Being on the first graph means f(x) equals that y, and being on the second means g(x) does, so f(x) = g(x) there.

Is the solution the x or the y of the intersection?

The x. The y is the value both sides share at that point, not the input that makes them equal.

What if the graphs never cross?

Then no real x makes the two sides equal, and the equation has no real solution.

How do I get more decimal places than the graph shows?

Make a table of f(x) − g(x). Where it changes sign, a solution lies between. Shrink the step and repeat.

Does this work for exponential or absolute value equations?

Yes. The argument never uses the type of function, so it works for any two functions that can be graphed.

What to learn next

Systems of a Line and a Parabola or CircleHow to solve a system of one linear and one quadratic equation by substitution and by graphing, and why such a system has two, one or no solutions.

Standards alignment

This lesson covers the following Common Core State Standards for Mathematics.

CCSS.MATH.CONTENT.HSA.REI.D.11Reasoning with Equations and InequalitiesExplain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.

For teachersPrint problems onlyTick to print a blank worksheet; leave it clear to print the lesson with the full answer key.

Quick answer

What you'll learn

One equation, two graphs

Why the crossing points are the solutions

Narrowing with a table

Other kinds of functions

Worked examples

Common mistakes

Practice problems

Frequently asked questions

Why do intersection points solve f(x) = g(x)?

Is the solution the x or the y of the intersection?

What if the graphs never cross?

How do I get more decimal places than the graph shows?

Does this work for exponential or absolute value equations?

What to learn next

Standards alignment

More Algebra 1 lessons