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Determine Number of Solutions for a Linear Equation

Students solve linear equations and recognize when the result is a unique solution (one value of x), no solution (a false statement like 3 = 5), or infinitely many solutions (a true statement like 0 = 0). They describe the structure of equations that lead to each case.

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Determine Number of Solutions for a Linear Equation

Grade 8 · Linear Equations in One Variable · question 1 of 3

0 of 20

20 rungs. A star at the top.

2x + 5 = 2x + 6 has no solution. Which of these equations has no solution too?

How it is taught

0:00 of 3:32

A linear equation can have one solution, none, or infinitely many.

A false number sentence means no solution.

Read it: How many solutions

7(x + 1) is 7x + 7. Against 7x + 7 every x works; against 7x + 9 none does; against 8x + 7 exactly one does, x = 0.

  1. Simplify each side: multiply out any parentheses.
  2. Compare the x-terms, then the numbers.
  3. Different x-terms: exactly one. Same x-terms and numbers: infinitely many. Same x-terms, different numbers: none.

When the x-terms differ, the two sides change at different rates, so they are equal at exactly one x.

When both sides are the same, every x makes them equal: infinitely many solutions.

When only the numbers differ, the sides stay the same distance apart, so no x works.

Simplify both sides; then the x-terms and the numbers decide: one, none or infinitely many.

Seeing the same 7 on both sides of 7x + 7 = 8x + 7 and calling it every x.

Compare the x-terms first: 7x and 8x differ, so exactly one x works, x = 0.

How many solutions does 9(x + 3) = 9x + 29 have? — no solution. 9(x + 3) = 9x + 29 is 9x + 27 = 9x + 29: the same x-terms but 27 and 29 differ, so no x works.

3(x + 2) = 3x + 6

Linear equations don't always have just one answer. Some have none, and some have infinitely many. Let's solve one and find out.

Here's our equation: three times the quantity x plus two, equals three x plus six.

First, we distribute the three across x plus two.

Three times x is three x, and three times two is six — so the left side becomes three x plus six.

Look closely — both sides are exactly the same!

Now subtract three x from both sides to simplify.

The x terms cancel, and we're left with six equals six.

Six equals six is always true — for every value of x!

Here's the rule: a true statement means infinitely many solutions, a false one means no solution, and a single value of x means just one solution.

So this equation has infinitely many solutions. Any number for x makes it true. Nice work!

How many solutions?One, none, or infinitely many?3(x + 2) = 3x + 63(x + 2) = 3x + 6distribute the 33x + 6 = 3x + 6Both sides are identical!− 3x− 3x6 = 6Always true — for every x ✓x = 4 → one solution5 = 7 → no solution6 = 6 → infinitely manyANSWER∞solutions
Grade 8 · One, none, or infinitely many?

What you see

While they climb, your page writes one true line to ask about tonight, and a short note each Sunday.

MayaGrade 3 · Mathematics
Did you know?

A chessboard has 64 squares — share them out between the 8 rows and each row holds exactly 8.

Today’s goal
1 star — one skill mastered, every rung of its ladder climbed
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Divide within 100

An example student.

MayaGrade 3 · Mathematics

Maya earned 2 stars this week — Interpret multiplication as equal groups, which took 6 answers over 3 days before its star landed 3 days ago and Multiply by 6, 7, 8 and 9. Next up: Divide within 100.

An example week.

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Determine Number of Solutions for a Linear Equation — Grade 8 Mathematics