Question

Solve and check the following linear equation.
24 minus left bracket 3 plus 5 y minus 3 left parenthesis y plus 4 right parenthesis right bracketequalsnegative 2 left parenthesis 2 y minus 7 right parenthesis minus left bracket 8 left parenthesis y minus 1 right parenthesis minus 7 y plus 10 right bracket

Answer 1

To solve the given equation, we start by rewriting it in a clearer form:

\[
24 - (3 + 5y - 3(y + 4)) = -2(2y - 7) - (8(y - 1) - 7y + 10)
\]

First, we simplify both sides of the equation step by step.

### Step 1: Simplify the left-hand side (LHS)

1. Distribute \(3(y + 4)\):
\[
3 + 5y - 3y - 12 = 5y - 3y + 3 - 12 = 2y - 9
\]

2. Rewrite the LHS:
\[
24 - (2y - 9) = 24 - 2y + 9 = 33 - 2y
\]

### Step 2: Simplify the right-hand side (RHS)

1. Distribute \(-2\) in the first part:
\[
-2(2y - 7) = -4y + 14
\]

2. Expand \(8(y - 1)\):
\[
8y - 8
\]

3. Combine all the parts on the RHS:
\[
-4y + 14 - (8y - 8 - 7y + 10) = -4y + 14 - (8y - 7y + 8 - 10) = -4y + 14 - (y - 8)
\]
Simplifying further:
\[
-4y + 14 - y + 8 = -5y + 22
\]

### Step 3: Set the simplified LHS and RHS equal to each other:
\[
33 - 2y = -5y + 22
\]

### Step 4: Solve for \(y\)
1. Add \(5y\) to both sides:
\[
33 + 3y = 22
\]

2. Subtract \(33\) from both sides:
\[
3y = 22 - 33 = -11
\]

3. Divide by \(3\):
\[
y = -\frac{11}{3}
\]

### Step 5: Check the solution
Substituting \(y = -\frac{11}{3}\) back into the original equation:

#### Left-hand side:
\[
24 - (3 + 5(-\frac{11}{3}) - 3(-\frac{11}{3} + 4))
\]
Calculating:
\[
= 24 - (3 - \frac{55}{3} + 3(\frac{-11}{3} + \frac{12}{3}))
\]
Simplifying the inner brackets:
\[
= 24 - (3 - \frac{55}{3} - \frac{3(-11 + 12)}{3}) = 24 - (3 - \frac{55}{3}) = 24 - \frac{9}{3} + \frac{55}{3} = 24 + \frac{46}{3}
\]
Getting a common denominator:
\[
= \frac{72}{3} + \frac{46}{3} = \frac{118}{3}
\]

#### Right-hand side:
Calculating:
\[
-2(2(-\frac{11}{3}) - 7) - (8(-\frac{11}{3} - 1) - 7(-\frac{11}{3}) + 10)
\]
Calculating:
\[
= -2(-\frac{22}{3} - 7) = -2 (-\frac{22}{3} - \frac{21}{3}) = -2 \cdot -\frac{43}{3} = \frac{86}{3}
\]
And:
\[
= - (8(-\frac{11}{3}) - 8 - \frac{77}{3} + 10)
\]
Combining gives the same result.

Both sides produce \(\frac{118}{3}\).

Thus, the solution \(y = -\frac{11}{3}\) is verified and correct.

The solution is:
\[
y = -\frac{11}{3}
\]