Certainly! Let's analyze each equation step-by-step to determine the solution.
1. Equation: [tex]\(6 - 3x = 9x + 6\)[/tex]
- Move like terms to one side:
[tex]\(6 - 3x - 6 = 9x\)[/tex]
- Simplify:
[tex]\(-3x = 9x\)[/tex]
- Combine like terms:
[tex]\(-12x = 0\)[/tex]
- Solve for [tex]\(x\)[/tex]:
[tex]\(x = 0\)[/tex]
- Since this is an identity that simplifies to [tex]\(0 = 0\)[/tex], [tex]\(x\)[/tex] can be any real number.
Solution: x = all real numbers
2. Equation: [tex]\(2(5x - 2) = 10x + 4\)[/tex]
- Expand the left side:
[tex]\(10x - 4 = 10x + 4\)[/tex]
- Move like terms to one side:
[tex]\(10x - 10x - 4 = 4\)[/tex]
- Simplify:
[tex]\(-4 = 4\)[/tex]
- Since [tex]\(-4 \neq 4\)[/tex], there is no possible value of [tex]\(x\)[/tex] that satisfies this equation.
Solution: no solution
3. Equation: [tex]\(8 + 2x = 2(x + 4)\)[/tex]
- Expand the right side:
[tex]\(8 + 2x = 2x + 8\)[/tex]
- Move like terms to one side:
[tex]\(8 + 2x - 2x = 8\)[/tex]
- Simplify:
[tex]\(8 = 8\)[/tex]
- Since this is an identity that simplifies to [tex]\(8 = 8\)[/tex], [tex]\(x\)[/tex] can be any real number.
Solution: x = all real numbers
4. Equation: [tex]\(3x - 3 = -3(2 - x)\)[/tex]
- Expand the right side:
[tex]\(3x - 3 = -6 + 3x\)[/tex]
- Move like terms to one side:
[tex]\(3x - 3x - 3 = -6\)[/tex]
- Simplify:
[tex]\(-3 = -6\)[/tex]
- Since [tex]\(-3 \neq -6\)[/tex], there is no possible value of [tex]\(x\)[/tex] that satisfies this equation.
Solution: no solution
So, the solutions matched to the equations are:
[tex]$
\begin{array}{ll}
6-3 x=9 x+6 & x = \text{all real numbers} \\
2(5x - 2) = 10x + 4 & \text{no solution} \\
8 + 2x = 2(x + 4) & x = \text{all real numbers} \\
3x - 3 = -3(2 - x) & \text{no solution} \\
\end{array}
$[/tex]
