Before you read

Answer first — see what you already know.

A cell phone company models two plans with the equations $5 x + 2$ and $5 x + 2$ . How many values of $x$ make both plans equal in cost?

A
No solution
B
Exactly one solution
C
Exactly two solutions
D
Infinitely many solutions

Check your answer

Answer:

Before you read

Answer first — see what you already know.

Two shipping options have costs modeled by $3 x + 7$ and $3 x - 1$ . How many solutions does the equation $3 x + 7 = 3 x - 1$ have?

A
No solution
B
Exactly one solution
C
$x = 0$
D
Infinitely many solutions

Check your answer

Answer:

Theory

Three Possible Outcomes

When you solve a linear equation, one of three things can happen:

1. One solution — You get a statement like $x = 5$ . Most equations work this way.

Intersecting lines: one solution

2. No solution — You get a false statement like $0 = 7$ or $3 = -1$ . This means no value of $x$ can ever make the equation true. We call this a contradiction.

Parallel lines: no solution (same slope, different intercepts)

3. Infinitely many solutions — You get a true statement like $0 = 0$ or $5 = 5$ . This means every value of $x$ works. We call this an identity.

The key is to simplify the equation completely and see what's left.

Theory

How to Recognize Each Case

After simplifying, look at what happens to the $x$ -terms:

What you get	Meaning	Example
$x =$ some number	One solution	$3x = 12$ $\to$ $x = 4$
A false number statement	No solution	$0 = 7$
A true number statement	Infinitely many	$0 = 0$

The trick: if the $x$ -terms cancel out on both sides, you're left with either a contradiction or an identity.

Example 1

How many solutions does $2(x + 3) = 2x + 6$ have?

Distribute: $2x + 6 = 2x + 6$

Subtract $2x$ from both sides: $6 = 6$

This is always true! Infinitely many solutions.

Any value of $x$ satisfies this equation.

Infinitely many solutions

Example 2

How many solutions does $3x + 5 = 3x - 2$ have?

Subtract $3x$ from both sides: $5 = -2$

This is never true! No solution.

No value of $x$ can make $5$ equal $-2$ .

No solution

Example 3

How many solutions does $4x + 1 = 2x + 9$ have?

Subtract $2x$ : $2x + 1 = 9$

Subtract $1$ : $2x = 8$

Divide by $2$ : $x = 4$

Exactly one solution: $x = 4$ .

One solution:

x = 4

Continue this lesson in the app

3 more sections including examples, practice problems, and step-by-step solutions.

Try NovaMath Free

Ready to ace the SAT?

Try NovaMath free — AI tutoring, 115 lessons, 2,400+ exercises.

Try NovaMath Free