Question

classify each equation as having no solution, one solution, or infinitely many solutions
no solution one solution infinitely many solutions
-4x + 4x = 0 -x = 7x - 8x 5 - x = -x x + 4 = x + 3 9 = 3(5x - 2) 6x - (3x + 8) = 18

Explanation:

Step1: Simplify each equation

• For $-4x + 4x=0$, the left - hand side simplifies to $0 = 0$.
• For $-x=7x - 8x$, simplify the right - hand side: $7x-8x=-x$, so $-x=-x$.
• For $5 - x=-x$, add $x$ to both sides, we get $5=0$, which is false.
• For $x + 4=x + 3$, subtract $x$ from both sides, we get $4 = 3$, which is false.
• For $9 = 3(5x-2)$, first distribute on the right - hand side: $9=15x-6$. Then add 6 to both sides: $15 = 15x$, and $x = 1$.
• For $6x-(3x + 8)=18$, first distribute the negative sign: $6x-3x-8=18$, then combine like terms: $3x-8=18$, add 8 to both sides: $3x=26$, and $x=\frac{26}{3}$.

Step2: Classify the equations

• Equations that simplify to a true statement for all $x$ (like $0 = 0$ and $-x=-x$) have infinitely many solutions.
• Equations that simplify to a false statement (like $5 = 0$ and $4 = 3$) have no solution.
• Equations that can be solved for a single value of $x$ (like $x = 1$ and $x=\frac{26}{3}$) have one solution.

Answer:

No Solution: $5 - x=-x$, $x + 4=x + 3$
One Solution: $9 = 3(5x-2)$, $6x-(3x + 8)=18$
Infinitely Many Solutions: $-4x + 4x=0$, $-x=7x - 8x$