Question

solve and check the following linear equation.
21 - 2 + 7y - 4(y + 2) = -4(4y - 7) - 9(y - 1) - 17y + 20
what is the solution? select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. the solution set is {}

b. the solution set is {x | x is a real number}

c. the solution set is ∅

Explanation:

Step1: Simplify both sides by distributing

Left side: \(21 - [2 + 7y - 4(y + 2)] = 21 - [2 + 7y - 4y - 8] = 21 - [3y - 6]\)
Right side: \(-4(4y - 7) - [9(y - 1) - 17y + 20] = -16y + 28 - [9y - 9 - 17y + 20] = -16y + 28 - [-8y + 11]\)

Step2: Remove brackets

Left side: \(21 - 3y + 6 = 27 - 3y\)
Right side: \(-16y + 28 + 8y - 11 = -8y + 17\)

Step3: Solve for y

\(27 - 3y = -8y + 17\)
Add \(8y\) to both sides: \(27 + 5y = 17\)
Subtract 27: \(5y = -10\)
Divide by 5: \(y = -2\)

Step4: Check the solution

Substitute \(y = -2\) into original equation:
Left: \(21 - [2 + 7(-2) - 4(-2 + 2)] = 21 - [2 -14 - 0] = 21 - (-12) = 33\)
Right: \(-4(4(-2) -7) - [9(-2 -1) -17(-2) +20] = -4(-8 -7) - [-27 +34 +20] = -4(-15) -27 = 60 -27 = 33\)
Both sides equal, so \(y = -2\) is valid.

Answer:

A. The solution set is \(\{-2\}\)