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Special Equations

Special Equations

One of three things will happen when solving a linear equation:

  • The equation will have exactly one solution.
  • The equation will have infinitely many solutions.
  • The equation will have no solution.

Equations with infinitely many solutions or no solution are called special equations.

You have learned how to solve equations with exactly one solution in the previous lesson. When an equation has one solution, only one real number satisfies the equation.

Now let’s learn about special equations and how to solve them.

Infinitely Many Solutions (All Real Numbers)

When an equation has infinitely many solutions, all real numbers, any value for the variable satisfies the equation. This means that you will always get a true statement when you substitute in any value for the variable. Consider the equation, 8(x + 3) = 4(2x + 6). Let’s see what we get when we substitute in the values –5, 0, 2, and 7 for the variable x.

For x = -5, we get:

  • 8(x + 3) = 4(2x + 6)
  • 8(-5 + 3) = 4(2(-5) + 6)
  • 8(-2) = 4(-10 + 6)
  • -16 = 4(-4)
  • -16 = -16
  • True Statement

For x = 0, we get:

  • 8(x + 3) = 4(2x + 6)
  • 8(0 + 3) = 4(2(0) + 6)
  • 8(3) = 4(0 + 6)
  • 24 = 4(6)
  • 24 = 24
  • True Statement

For x = 2, we get:

  • 8(x + 3) = 4(2x + 6)
  • 8(2 + 3) = 4(2(2) + 6)
  • 8(5) = 4(4 + 6)
  • 40 = 4(10)
  • 40 = 40
  • True Statement

For x = 7, we get:

  • 8(x + 3) = 4(2x + 6)
  • 8(7 + 3) = 4(2(7) + 6)
  • 8(10) = 4(14 + 6)
  • 80 = 4(20)
  • 80 = 80
  • True Statement

Each of the values, -5, 0, 2, and 7, gives a true statement. Therefore the values, -5, 2, 0, and 7 are solutions to the equation.

If you substitute any real number in the equation, you will always get a true statement. So, the equation, 8(x + 3) = 4(2x + 6), has infinitely many solutions, all real numbers.

No Solution

When an equation has no solution, no value for the variable satisfies the equation. This means that you will always get a false statement when you substitute in any value for the variable. Consider the equation, 3(x + 2) = 3x + 2. Let’s see what we get when we substitute in the values –5, 0, 2, and 7 for the variable x.

For x = -5, we get:

  • 3(x + 2) = 3x + 2
  • 3(-5 + 2) = 3(-5) + 2
  • 3(-3) = -15 + 2
  • -9 = -13
  • False Statement

For x = 0, we get:

  • 3(x + 2) = 3x + 2
  • 3(0 + 2) = 3(0) + 2
  • 3(2) = 0 + 2
  • 6 = 2
  • False Statement

For x = 2, we get:

  • 3(x + 2) = 3x + 2
  • 3(2 + 2) = 3(2) + 2
  • 3(4) = 6 + 2
  • 12 = 8
  • False Statement

For x = 7, we get:

  • 3(x + 2) = 3x + 2
  • 3(7 + 2) = 3(7) + 2
  • 3(9) = 21 + 2
  • 27 = 24
  • False Statement

Each of the values, -5, 0, 2, and 7, gives a false statement. Therefore the values, -5, 2, 0, and 7 are not solutions to the equation.

If you substitute any real number in the equation, you will always get a false statement. So, the equation, 3(x + 2) = 3x + 2, has no solution.

Types of Solutions

How do you know when a linear equation has one solution, infinitely many solutions, or no solution when you solve the equation?

One Solution: When you solve a linear equation and you can get an answer like x = -6, the equation has exactly one solution.

  • 4x + 5 = 2x – 7
  • 4x + 5 – 2x = 2x – 7 – 2x Subtract 2x from both sides
  • 2x + 5 = -7
  • 2x + 5 – 5 = -7 – 5 Subtract 5 from both sides
  • 2x = -12
  • (2x)/2 = (-12)/2 Divide both sides by 2
  • x = -6

Infinitely Many or All Real Numbers: When you solve a linear equation and you get an answer with the same number on both sides or the same variable on both sides like -4 = -4 or x = x, the equation has infinitely many solutions, all real numbers.

Infinitely Many Solutions Example #1:

  • 4(x + 1) – 8 = 4x – 4
  • 4x + 4 – 8 = 4x – 4
  • 4x – 4 = 4x – 4
  • –4x = –4x
  • -4 = -4

Infinitely Many Solutions Example #2:

  • 6(x + 3) – 1 = 6x + 17
  • 6x + 18 – 1 = 6x + 17
  • 6x + 17 = 6x + 17
  • 6x + 17 – 17 = 6x + 17 – 17 Subtract 17 from both sides
  • (6x)/6 = (6x)/6 Divide both sides by 6
  • x = x

No Solution: When you solve a linear equation and you get no answer or an answer that isn't true, then the equation has no solution. For example, if you get the answer 2 = 1, the equation has no solution.

  • 5x – 2x + 2 = 3x + 1
  • 3x + 2 = 3x + 1
  • 3x + 2 – 3x= 3x + 1 – 3x Subtract 3x from both sides
  • 2 = 1

Example #1

Solve this problem and then identify the solution.

  • 2(x + 5) = 4x + 6
  • __blank + __blank = 4x + 6   Distribute.
  • 2x + 10 = 4x + 6
  • __blank + 10 = 6   Subtract 4x from each side.
  • −2x + 10 = 6
  • __blank = __blank   Subtract 10 from each side.
  • −2x = −4
  • x = __blank   Divide by −2.
  • x = 2

The solution to the problem above is:

  1. No solution
  2. Infinitely many solutions
  3. Exactly one solution

Answer: c. Exactly one solution

Solve Equations in One Variable that May or May Not Have a Solution Example #2

Let’s look at an example that may or may not have a solution.

Open Solve Equations in One Variable that May or May Not Have a Solution in a new tab

Example #3

Solve this problem and then identify the solution.

  • 2y + 4 = 2(y + 2)
  • 2y + 4 = __blank + __blank   Distribute.
  • 2y + 4 = 2y + 4
  • __blank = __blank   Subtract 2y from each side.
  • 4 = 4

The solution to the problem above is:

  1. No solution
  2. Infinitely many solutions
  3. Exactly one solution

Answer: b. Infinitely many solutions

Example #4

Let’s try another example.

9m − 4 = −3m + 5 + 12m

Combine like terms: 9m − 4 = 9m + 5

Subtract 9m from each side: −4 = 5

Notice that the variables are gone from each side and that we get −4 = 5, which is not a true statement. This means that we have No Solution. These are the lines that never cross.

Example #5

Solve this problem and then identify the solution.

  • 3x + 4 = 5x − 5 − 2x
  • 3x + 4 = __blank − 5   Combine like term.
  • 3x + 4 = 3x − 5
  • __blank = __blank   Subtract variables from each side.
  • 4 = −5

The solution to the problem above is:

  1. No solution
  2. Infinitely many solutions
  3. Exactly one solution

Answer: a. No solution

Reminder!

We can get three types of answers when we solve an equation:

  1. One solution, x = ?
  2. Infinitely many solutions
  3. No solution