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How can i tell if an equation like 2x + 3 = 2x + 7 has one solution, no solutions, or infinite solutions?

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To determine whether an equation like \(2x + 3 = 2x + 7\) has one solution, no solutions, or infinite solutions, you can follow these steps:

  1. Rearrange the equation: Start by simplifying both sides of the equation to see if you can isolate \(x\).

    \[ 2x + 3 = 2x + 7 \]

  2. Subtract \(2x\) from both sides:

    \[ 3 = 7 \]

  3. Analyze the result:

    • If you end up with a true statement (like \(5 = 5\)), it means that the original equation is true for all values of \(x\), indicating infinite solutions.
    • If you end up with a false statement (like \(3 = 7\)), it indicates that there are no solutions because no value of \(x\) can satisfy the equation.
    • If you had been able to isolate \(x\) and found a specific value (like \(x = 2\)), then there would be one solution.

In the case of \(2x + 3 = 2x + 7\), since we found \(3 = 7\), which is false, we conclude that there are no solutions to this equation.

What equations would fall under that

Equations that would lead to a false statement when simplified (indicating no solutions) usually have the same variable terms on both sides but different constant terms. Here are some examples:

  1. Example 1: \[ 4x + 2 = 4x + 5 \] Simplifying gives: \[ 2 = 5 \quad (\text{false}) \]

  2. Example 2: \[ 3x - 1 = 3x + 4 \] Simplifying gives: \[ -1 = 4 \quad (\text{false}) \]

  3. Example 3: \[ 5x + 8 = 5x + 10 \] Simplifying gives: \[ 8 = 10 \quad (\text{false}) \]

  4. Example 4: \[ 7 - 2y = 7 - 3y \] Simplifying gives: \[ 2y = 3 \quad\Rightarrow\quad 7 = 7 + 2 \quad\Rightarrow\quad 7 = 7(1) \quad (\text{false, and leads to } 0 = 1) \]

Equations with Infinite Solutions

Equations that yield a true statement after simplification often have the same variable terms and the same constant terms, leading to infinite solutions:

  1. Example 1: \[ 2x + 4 = 2x + 4 \] This simplifies to \(4 = 4\), which is always true.

  2. Example 2: \[ 3(x + 1) = 3x + 3 \] This simplifies to \(3 = 3\), which is always true.

  3. Example 3: \[ 5x + 6 - 2 = 5x + 4 \] This simplifies to \(4 = 4\), which is always true.

  4. Example 4: \[ 8 = 3 + 5 \] This simplifies to \(8 = 8\), which is always true.

Equations with One Solution

Equations that result in a specific value of \(x\) once simplified yield exactly one solution:

  1. Example 1: \[ x + 3 = 7 \] This simplifies to \(x = 4\).

  2. Example 2: \[ 2x - 4 = 10 \] This simplifies to \(2x = 14\) and \(x = 7\).

  3. Example 3: \[ 3y + 2 = 11 \] This simplifies to \(3y = 9\) and \(y = 3\).

  4. Example 4: \[ 4z + 6 = 18 \] This simplifies to \(4z = 12\) and \(z = 3\).

By understanding these types of equations, you can categorize them based on their solutions.

what about 4x+22=8x+10

Let’s analyze the equation \(4x + 22 = 8x + 10\) to determine whether it has one solution, no solutions, or infinite solutions.

  1. Rearrange the equation: Start by simplifying both sides to get all terms involving \(x\) on one side and constant terms on the other side.

    \[ 4x + 22 = 8x + 10 \]

  2. Subtract \(4x\) from both sides: \[ 22 = 8x - 4x + 10 \] This simplifies to: \[ 22 = 4x + 10 \]

  3. Next, subtract \(10\) from both sides: \[ 22 - 10 = 4x \] This simplifies to: \[ 12 = 4x \]

  4. Now, divide both sides by \(4\): \[ x = \frac{12}{4} \] Thus, \[ x = 3 \]

Conclusion

The equation \(4x + 22 = 8x + 10\) has one solution, which is \(x = 3\).

This is an example of an equation with one unique solution, as we isolated \(x\) and found a specific value for it.

3 (4x + 8) = 4 (3x + 6)?

Let's analyze the equation \(3(4x + 8) = 4(3x + 6)\) step by step to determine whether it has one solution, no solutions, or infinite solutions.

  1. Distribute both sides of the equation:

    \[ 3(4x + 8) = 4(3x + 6) \]

    • Left side: \[ 3 \cdot 4x + 3 \cdot 8 = 12x + 24 \]

    • Right side: \[ 4 \cdot 3x + 4 \cdot 6 = 12x + 24 \]

    So the equation becomes: \[ 12x + 24 = 12x + 24 \]

  2. Reorganize the equation:

    Now, let's see what we have: \[ 12x + 24 = 12x + 24 \]

  3. Subtract \(12x\) from both sides:

    \[ 24 = 24 \]

Conclusion

The statement \(24 = 24\) is always true for any value of \(x\).

This means that the original equation has infinite solutions; any value of \(x\) will satisfy the equation.