Question

The equation shown has a missing value. Answer parts a through c. - 2(2x- ) + 1 = 17 - 4x a. For what missing value is the equation an identity? b. For what missing value(s), if any, does the equation have exactly one solution? c. For what missing value(s), if any, does the equation have no solution? a. The equation is an identity when the missing value is 8. b. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. O A. The equation has exactly one solution when the missing value is anything except (Use a comma to separate answers as needed.) B. The equation has exactly one solution when the missing value is (Use a comma to separate answers as needed.) C. The equation never has exactly one solution regardless of the missing value. c. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. O A. The equation has no solution when the missing value is anything except (Use a comma to separate answers as needed.) B. The equation has no solution when the missing value is . (Use a comma to separate answers as needed.) C. The equation always has at least one solution, regardless of the missing value.​

Answer 1

Equations can have one solution, more than one solution or none at all.

• The equation is an identity when the missing value is 8
• The equation cannot have exactly one solution
• The equation has exactly one solution when the missing value is anything except 8

The equation is given as:

$2(2x - []) + 1 = 17 - 4x$

Represent the blank with y.

So, we have:

$-2(2x - y) + 1 = 17 - 4x$

(a) The missing value when the equation is an identity equation

To do this, we simply solve for y in $-2(2x - y) + 1 = 17 - 4x$

Open brackets

$-4x +2y + 1 = 17 - 4x$

Collect like terms

$2y = 17 - 1 + 4x - 4x$

$2y = 16$

Divide both sides by 2

$y = 8$

Hence, the equation is an identity when the missing value is 8

(b) The missing value when the equation has one solution

In (a), we have: $y = 8$

For the equation to have one solution, variable x must be on either sides of the equation.

i.e. $nx + y = 8$ or $y = 8 + nx$, where $n \ne 0$

Since x has been eliminated, then the equation can not have one solution.

(c) The missing value when the equation has no solution

In (a), we have: $y = 8$

This means that the equation will have no solution when the missing value is not 8

i.e.

$y \ne 8$

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