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Math  /  Algebra

Question效, Find the missing number so that the equation has no solutions. □\square x+6=5x−16x+6=5 x-16 Submit

Studdy Solution

STEP 1

1. We are given an equation of the form □x+6=5x−16 \square x + 6 = 5x - 16 .
2. The goal is to find a value for the blank (denoted as □\square) such that the equation has no solutions.
3. An equation has no solutions when it results in a contradiction, such as a false statement.

STEP 2

1. Identify the structure of the equation.
2. Rearrange the equation to isolate terms involving x x .
3. Determine the condition for no solutions.
4. Solve for the missing number that satisfies the no-solution condition.

STEP 3

Identify the structure of the equation. The equation is:
□x+6=5x−16 \square x + 6 = 5x - 16
Here, □\square represents the coefficient of x x on the left side of the equation.

STEP 4

Rearrange the equation to isolate terms involving x x . Move all terms involving x x to one side and constant terms to the other side:
□x−5x=−16−6 \square x - 5x = -16 - 6
This simplifies to:
(□−5)x=−22 (\square - 5)x = -22

STEP 5

Determine the condition for no solutions. For the equation to have no solutions, the coefficient of x x must be zero, leading to a contradiction with the constants:
(□−5)x=−22 (\square - 5)x = -22
For no solutions, □−5\square - 5 must equal 0, which would make the equation:
0⋅x=−22 0 \cdot x = -22
This is a contradiction because 0 cannot equal -22.

STEP 6

Solve for the missing number that satisfies the no-solution condition. Set □−5=0\square - 5 = 0 and solve for □\square:
□−5=0 \square - 5 = 0 □=5 \square = 5
The missing number is 5 \boxed{5} .

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