Question

For the equation 2(4+x)+(13+x)=3x+k which value of k will create an equation with no solutions. .A. x B. 3x .C 15 D. 21

Answer 1

Answer: choice C) 15

Simplify the left side to get

2(4+x)+(13+x)

2(4)+2(x) +13+x

8+2x+13+x

3x+21

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So the original equation

2(4+x)+(13+x) = 3x+k

turns into

3x+21 = 3x+k

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Subtract 3x from both sides

3x+21 = 3x+k

3x+21-3x = 3x+k-3x

21 = k

k = 21

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If k = 21, then the original equation will have infinitely many solutions. This is because we will end up with 3x+21 on both sides, leading to 0 = 0 after getting everything to one side. This is a true equation no matter what x happens to be.

If k is some fixed number other than 21, then there will be no solutions. This equation is inconsistent (one side says one thing, the other side says something different). If k = 15, then

3x+21 = 3x+k

3x+21 = 3x+15

21 = 15 .... subtract 3x from both sides

The last equation is false, so there are no solutions here.

note: if you replace k with a variable term, then there will be exactly one solution.