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QuestionSolve for the missing value in the equation −2(2x−□)+1=17−4x-2(2x-\square)+1=17-4x that makes it an identity, has one solution, or no solution.

Studdy Solution

STEP 1

Assumptions1. The equation given is −(x−□)+1=17−4x-( x-\square)+1=17-4 x . The missing value is represented by □\square
3. An identity is an equation that is true for all values of x

STEP 2

First, we need to simplify the equation. Let's start by distributing the -2 on the left side of the equation.
−2(2x−□)+1=−4x+2□+1-2(2 x-\square)+1 = -4x +2\square +1

STEP 3

Now, we have the equation as−x+2□+1=17−x-x +2\square +1 =17 -x

STEP 4

For the equation to be an identity, the coefficients of xx on both sides should be equal and the constants on both sides should also be equal.Therefore, the coefficient of xx on both sides is -4, so they are already equal.Now, we need to find the value of □\square for which the constants on both sides are equal.So, we set 2□+1=172\square +1 =17.

STEP 5

olving the equation 2□+1=172\square +1 =17 for □\square gives2□=17−12\square =17 -1

STEP 6

implify the right side of the equation2□=162\square =16

STEP 7

Finally, solve for □\square by dividing both sides by2□=16/2\square =16 /2

STEP 8

Calculate the value of □\square□=8\square =8So, the equation is an identity when the missing value is8.
For parts b and c, we need to consider different scenarios.
b. The equation has exactly one solution when the coefficients of xx on both sides are not equal, which would make the equation a linear equation with one solution. However, in this case, the coefficients of xx on both sides are always equal (-4), regardless of the value of □\square. Therefore, there are no missing values for which the equation has exactly one solution.
c. The equation has no solution when the coefficients of xx on both sides are equal, but the constants are not equal. This would mean that the two expressions are parallel lines that never intersect. In this case, the coefficients of xx on both sides are always equal (-4), so we need to find the value of □\square for which 2□+1≠172\square +1 \neq17. Solving the inequality 2□+1≠172\square +1 \neq17 gives □≠8\square \neq8. Therefore, the equation has no solution for all values of □\square except8.

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