Questions: For each equation, choose the statement that describes its solution. If applicable, give the solution. 4(3+y)-y=6+3(y+2) No solution y= All real numbers are solutions 2(u+1)+7=3(u-2)+2u No solution u= All real numbers are solutions

For each equation, choose the statement that describes its solution. If applicable, give the solution.

4(3+y)-y=6+3(y+2)
No solution
y=
All real numbers are solutions

2(u+1)+7=3(u-2)+2u
No solution
u=
All real numbers are solutions

Transcript text: For each equation, choose the statement that describes its solu If applicable, give the solution. \[ 4(3+y)-y=6+3(y+2) \] No solution $y=$ $\square$ All real numbers are solutions \[ 2(u+1)+7=3(u-2)+2 u \] No solution $u=$ $\square$ All real numbers are solutions

Solution

Solution Steps

To solve each equation, we will simplify both sides and then isolate the variable to determine if there is a specific solution, no solution, or if all real numbers are solutions.

Equation 1: $4(3+y)-y=6+3(y+2)$
- Distribute and simplify both sides.
- Combine like terms.
- Solve for $y$.
Equation 2: $2(u+1)+7=3(u-2)+2u$
- Distribute and simplify both sides.
- Combine like terms.
- Solve for $u$.

Step 1: Simplify and Solve Equation 1

Given the equation:

\[ 4(3+y) - y = 6 + 3(y+2) \]

Distribute and simplify both sides: \[ 12 + 4y - y = 6 + 3y + 6 \] \[ 12 + 3y = 12 + 3y \]
Both sides are identical, indicating that the equation is true for all $ y $.

Step 2: Simplify and Solve Equation 2

Given the equation:

\[ 2(u+1) + 7 = 3(u-2) + 2u \]

Distribute and simplify both sides: \[ 2u + 2 + 7 = 3u - 6 + 2u \] \[ 2u + 9 = 5u - 6 \]
Rearrange to isolate $ u $: \[ 9 + 6 = 5u - 2u \] \[ 15 = 3u \] \[ u = 5 \]