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How many solutions? QuickChek

Which of the following is example of an equation with no solution
3X +4 equals 3X +4
4X +3 equals 3X +3
3X +4 equals 3X +3
3X +4 = 4X +3

9 answers

To determine which of the given equations has no solution, let's analyze each one:

  1. \(3X + 4 = 3X + 4\)
    This equation is true for all values of \(X\) because both sides are identical. Therefore, it has infinitely many solutions.

  2. \(4X + 3 = 3X + 3\)
    If we subtract \(3X\) from both sides, we get: \[ 4X - 3X + 3 = 3 \] Simplifying yields: \[ X + 3 = 3 \implies X = 0 \] This equation has exactly one solution (X = 0).

  3. \(3X + 4 = 3X + 3\)
    Subtracting \(3X\) from both sides gives us: \[ 4 = 3 \] This statement is false, indicating there are no values of \(X\) that satisfy this equation. Therefore, it has no solution.

  4. \(3X + 4 = 4X + 3\)
    Rearranging gives us: \[ 3X + 4 - 4X = 3 \implies -X + 4 = 3 \] Simplifying yields: \[ -X = -1 \implies X = 1 \] This equation has exactly one solution (X = 1).

In summary, the equation \(3X + 4 = 3X + 3\) is the example of an equation with no solution.

Which of the following is an example of an equation with exactly one solution?
3X +4 = 4X +3
4X +3 = 4X +3
3X +4 = 3X +3
3X +4 = 3X +4

To identify which of the given equations has exactly one solution, let's examine each one:

  1. \(3X + 4 = 4X + 3\)
    Rearranging gives: \[ 3X + 4 - 4X = 3 \implies -X + 4 = 3 \] Simplifying this: \[ -X = -1 \implies X = 1 \] This equation has exactly one solution: \(X = 1\).

  2. \(4X + 3 = 4X + 3\)
    This equation is an identity; both sides are exactly the same. Thus, it has infinitely many solutions.

  3. \(3X + 4 = 3X + 3\)
    Subtracting \(3X\) from both sides gets us: \[ 4 = 3 \] This is a contradiction, indicating that there are no solutions.

  4. \(3X + 4 = 3X + 4\)
    Similar to the second equation, this is also an identity and has infinitely many solutions.

In conclusion, the equation \(3X + 4 = 4X + 3\) has exactly one solution, which is \(X = 1\).

Solve X determine if the equation has one non-or infinite solutions.
3 (X +2) -6 = 3X
The equation has no solution
The equation has infinite solutions
The equation has one solution X = 3
The equation has one solution X = -3

Let's solve the equation step by step to determine how many solutions it has.

Given equation: \[ 3(X + 2) - 6 = 3X \]

  1. Distribute the 3 on the left side: \[ 3X + 6 - 6 = 3X \] This simplifies to: \[ 3X = 3X \]

  2. Now, we can subtract \(3X\) from both sides: \[ 3X - 3X = 3X - 3X \implies 0 = 0 \]

This equation \(0 = 0\) is always true, meaning that all values of \(X\) satisfy the equation.

Conclusion:

Since the equation is true for all values of \(X\), it has infinite solutions.

So the correct answer is:

  • The equation has infinite solutions.
Solve for X determine if the equation has one none or infinite solutions
4X = 3 (X +5) -15
The equation has one solution X = -10
The equation has no solution
The equation has infinite solutions
The equation has one solution = 0

Let's solve the equation step by step to determine how many solutions it has.

Given equation: \[ 4X = 3(X + 5) - 15 \]

  1. Expand the right side: \[ 4X = 3X + 15 - 15 \] Simplifying the right side, we get: \[ 4X = 3X + 0 \implies 4X = 3X \]

  2. Subtract \(3X\) from both sides: \[ 4X - 3X = 3X - 3X \implies X = 0 \]

Conclusion:

The equation has one solution:

  • \(X = 0\).

So, the correct answer from your options is:

  • The equation has one solution \(X = 0\).
Solve for extra to determine if the equation is one none or infinite solutions
5(2x+2)-7X = 3X +6
The equation has infinite solutions
The equation has one solution, X = 10
The equation has no solution
The equation has one solution X = 6

Let's solve the equation step by step to determine how many solutions it has.

Given equation: \[ 5(2x + 2) - 7X = 3X + 6 \]

  1. Distribute the 5 on the left side: \[ 10x + 10 - 7x = 3x + 6 \]

  2. Combine like terms on the left side: \[ (10x - 7x) + 10 = 3x + 6 \] This simplifies to: \[ 3x + 10 = 3x + 6 \]

  3. Subtract \(3x\) from both sides: \[ 3x + 10 - 3x = 3x + 6 - 3x \implies 10 = 6 \]

This statement \(10 = 6\) is a contradiction, meaning there is no value of \(x\) that can satisfy the equation.

Conclusion:

The equation has no solution.

So, the correct answer from your options is:

  • The equation has no solution.