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Equations with Infinite Solutions and Equations with No Solution
> ... Math > Solving for a Variable > Equations with Infinite Solutions and Equations with No Solution

Introduction

In this lesson, you will learn about two different types of equations: equations with infinite solutions and equations with no solution.

This video illustrates the lesson material below. Watching the video is optional.

Infinite Solutions

There are three types of answers you can get when solving for a variable:

  • \(x=a\): where a represents all real numbers.
    • Example: \( x = 5.3\), \(x = -2\), \(x = 1.5\)
  • \(x =\) Infinitely Many Solutions: where x represents all real numbers or infinitely many solutions.
    • Example: \(3 = 3\), \(x =x \), \(-7 = -7\)
  • \(x =\) No Solution: no solution is when the statement is false.
    • Example: \(1 \neq 5\), \(0 \neq 4\), \(3 \neq -2\)

Not all equations will end with \(x =\) a specific number. Some equations may have infinitely many solutions and other equations may have no solution at all.

Example 1: Infinite Solutions
\begin{align*}2x + 3 &= 2x + 3 &\color{red}\small\text{Given equation}\\\\2x +3 \color{red}\mathbf{-3} &= 2x +3 \color{red}\mathbf{-3} &\color{red}\small\text{Subtract \(3\) from each side}\\\\ 2x &= 2x &\color{red}\small\text{Simplify each side}\\\\ \frac{2x}{\color{red}\mathbf{2}} &=\frac{2x}{\color{red}\mathbf{2}} &\color{red}\small\text{Divide each side by \(2\)}\\\\x &= x &\color{red}\small\text{Simplify each side}\\\\\end{align*}

This answer will always be true, no matter what value \(x\) might be. \(x\) can be any real number, and this would still be true; therefore, this equation has infinite solutions.

Example 2: Infinite Solutions
Simplify Example 1 another way:

\begin{align*}2x + 3 &= 2x + 3 &\color{red}\small\text{Given equation}\\\\ 2x +3 \color{red}\mathbf{-2x} &= 2x +3 \color{red}\mathbf{-2x} &\color{red}\small\text{Subtract \(2x\) from each side}\\\\ 3 &= 3 &\color{red}\small\text{Simplify each side}\\\end{align*}

When you solve an equation and get the same answer on each side of the equals sign, such as \(3\) equals \(3\), or \(x\) equals \(x\), all real numbers are the solution, or the equation has infinite solutions.

Example 3: No Solutions
\begin{align*}2x + 1 &= 2x -5 &\color{red}\small\text{Given equation}\\\\2x +1 \color{red}\mathbf{-2x} &= 2x -5 \color{red}\mathbf{-2x} &\color{red}\small\text{Subtract 2x from both sides}\\\\1 &= -5 &\color{red}\small\text{Simplify each side} \end{align*}

The two \(x\)'s from each side will cancel out, leaving \(1=-5\). This is a false statement. 1 is not equal to -5. Therefore, no matter what value you put into this equation for \(x\), it will never be true. This equation has no solutions.

Example 4: No Solutions
Simplify Example 3 another way:

\begin{align*}2x + 1 &= 2x - 5 &\color{red}\small\text{Given equation}\\\\2x + 1 \color{red}\mathbf{-1} &= 2x - 5 \color{red}\mathbf{-1} &\color{red}\small\text{Subtract 1 from each side}\\\\2x &= 2x -6 &\color{red}\small\text{Simplify each side}\\\\ 2x\color{red}\mathbf{-2x} &=2x - 6\color{red}\mathbf{-2x} &\color{red}\small\text{Subtract 2x from each side}\\\\ 0&= -6 &\color{red}\small\text{Simplify each side}\\\\\end{align*}

Again, the two \(x\)'s cancel each other out, leaving \(0=6\), which is not true. When you solve an equation and come up with a false statement like this one, there is no solution.

Things to Remember


  • Anytime you solve an equation and get the same result on each side of the equal sign, or a true statement, the problem has infinite solutions and all real numbers are the solution.
  • Anytime you solve an equation and find a false statement at the end, it means there are no solutions.

Practice Problems

Identify whether there is one solution, infinitely many solutions, or no solution:
  1. \(-9{\text{M}} {-} 4 = -9{\text{M}} - 4\) (
    Solution
    x
    Solution: Infinitely many solutions
    Details:
    In this example, the first thing to do is combine like terms. This means combining the terms with the variable \(M\) with each other and combining the terms without a variable together.

    Note: There are two versions or ways to solve this equation. Either one is acceptable. You do not have to do both.

    First version: Combine terms with variable M first

    There is currently a \(−9M\) on the right-hand side of the equation. Remove it from the right-hand side and combine it with the left-hand side by adding \(+9M\) to both sides of the equation.

    The equation negative nine M minus four equals negative nine M minus four. There is a vertical dashed line through the equal sign separating the left and right sides of the equation. There is an arrow pointing to the negative 9 M on the right side. Under the negative nine M on both sides is a positive nine M, showing that 9 M will be added to both sides of the equation. 

    On the right-hand side:

    \(-9M + 9M = 0\) leaving just \(-4\)

    The equation negative nine M minus four equals negative nine M minus four. There is a vertical dashed line through the equal sign separating the left and right sides of the equation. Under the negative nine M is a positive nine M being added to both sides. There is a horizontal line below this, showing that the addition is taking place. Below the line are the results of the operations, but only for the right-hand side which reads zero minus four. The zero is the result of negative 9M plus 9M. 

    On the left-hand side:

    \(-9M + 9M = 0\) leaving just \(-4\)

    The equation negative nine M minus four equals negative nine M minus four. There is a vertical dashed line through the equal sign separating the left and right sides of the equation. Under the negative nine M is a positive nine M being added to both sides. Below the previous material is a solid line showing that addition is taking place. Below the line are the results of the operations, which reads: zero minus 4 equals zero minus 4. 

    Because \(−9M+9M=0\), you are left with \(−4=−4\). This statement is always true, therefore, there are infinitely many solutions for the equation \(−9M−4=−9M−4\). This means that any value of M will still make this equation true

    The final solution: Infinitely many solutions.

    Second version: Combine terms without a variable first

    You want to combine the \(−4\) on the left-hand side of the equation with the \(−4\) on the right-hand side of the equation. To do this, add \(+4\) to both sides of the equation.

    The equation negative nine M minus four equals negative nine M minus four. There is a vertical dashed line through the equal sign separating the left and right sides of the equation. There is a positive four under the minus four on both sides of the equal sign. A blue arrow above the minus four and positive four indicates we are working with these terms. 

    On the left-hand side:

    \(−4+4=0\)

    On the right-hand side:

    \(−4+4=0\)

    The equation negative nine M minus four equals negative nine M minus four. There is a vertical dashed line through the equal sign separating the left and right sides of the equation. There is a positive four under the minus four on both sides of the equal sign. Below all of this is a horizontal line showing that addition is taking place, and beneath it is the solution which is: negative nine M plus zero equals negative nine M plus zero. The plus zero on both sides is in blue, indicating it is the part of the equation that changed from the original due to the addition of positive four to both sides. 

    This leaves you with \(−9M=−9M\). You can either stop here because you see both sides are equal to each other, which means that for any value of M the statement will be true, or you can keep solving for \(M\).

    To keep solving for \(M\), you need to multiply both sides by the multiplicative inverse of \(−9\). Multiply both sides by \(-\dfrac{1}{9}\).

    The equation (negative one-ninth) multiplied to (negative nine M) equals (negative one-ninth) multiplied to (negative nine M). The fractions one-ninth on both sides of the equal sign are green, indicating they are new. 

    \(\left (-\dfrac{1}{9} \right )\left ( -9 \right )=1\)

    This leaves \(1M=1M\).

    Three equations listed one above the other. The first equation is (negative one-ninth) multiplied to (negative nine M) equals (negative one-ninth) multiplied to (negative nine M). The fractions one-ninth on both sides of the equal sign are green. The equation below this is the same except the negative 9 is separated by parentheses from the M on both sides, which reads: (negative one-ninth) multiplied to (negative nine) multiplied to M equals (negative one-ninth) multiplied to (negative nine) multiplied to M. There is a bracket below the (negative one-ninth) multiplied to (negative nine) on both sides, showing the result of this multiplication equals 1 (because multiplying by the inverse equals 1). The third equation is 1 multiplied to M equals 1 multiplied to M. 

    \(M=M\) is always true for any value of \(M\).

    The final solution: Infinitely many solutions.
    )
  2. \(9 + 8{\text{T}} = 13{\text{T}} + 2\) (
    Solution
    x
    Solution: One solution
    )
  3. \(-4 + 2{\text{b}} = 2{\text{b}} - 9\) (
    Solution
    x
    Solution: No solution
    Details:
    Start by combining like terms.

    Combine the terms with the variable b by adding \(−2b\) to both sides of the equation.

    The equation negative four plus two b equals two b minus nine. There is a vertical dashed line through the equal sign separating the left and right sides of the equation. Under the two b on both sides is plus negative two b. The negative two b is in red indicating it is new. 

    Since \(2b+(−2b)=0\), you are left with \(−4\) on the left-hand side and \(−9\) on the right-hand side.

    The equation negative four plus two b equals two b minus nine. There is a vertical dashed line through the equal sign separating the left and right sides of the equation. Under the two b on both sides is plus negative two b. The negative two b is in red indicating it is new. Below all of this is a horizontal line showing that addition is being performed. Below it are the results of the operations above which reads: negative four plus zero equals zero minus nine. The zero on both sides is written in green to indicate it is the result of the addition above it. Under all of this is final new equation, which is: negative four equals negative nine. 

    But \(−4\) does not equal \(−9\).

    This means that no matter what values you put into this equation, it is not true.

    The final solution: No solution.
    )
  4. \(-7 + 7{\text{b}} + 18 = 3{\text{b}} + 3 - 4{\text{b}}\) (
    Solution
    x
    Solution: One solution
    )
  5. \(2{\text{x}} + 5 + {\text{x}} = -1 + 3{\text{x}} + 6\) (
    Solution
    x
    Solution: Infinitely many solutions
    )
  6. \(2(3{\text{X}} + 4) = 6{\text{X}} + 7\) (
    Video Solution
    x
    Solution: No solution
    Details:

     | Transcript)
  7. \(-4(4{\text{M}} {-} 3) = -16{\text{M}} + 12\) (
    Video Solution
    x
    Solution: Infinitely many solutions
    Details:

     | Transcript)
Solving Variables on Both Sides of the Equation
Substitute Values into an Equation and Solve for a Variable