Question

group the equations based on their number of solutions. no solution (true for no values) one solution (true for one value) infinitely many solutions (true for all values) $y\cdot(-6)\cdot(-3)=2\cdot y\cdot9$ $v + 2 = v - 2$ $2n = 2n$ $\frac{1}{2}+x = \frac{1}{3}+x$ $3(n + 1)=3n + 1$ $2n = n$ $2t + 6 = 2(t + 3)$ $7 - r = r - 7$

Explanation:

Step1: Simplify first equation

$y \cdot (-6) \cdot (-3) = 2 \cdot y \cdot 9$
$18y = 18y$

Step2: Simplify second equation

$v+2 = v-2$
$2 = -2$ (false)

Step3: Simplify third equation

$2n = 2n$
$0=0$ (always true)

Step4: Simplify fourth equation

$\frac{1}{2}+x = \frac{1}{3}+x$
$\frac{1}{2}=\frac{1}{3}$ (false)

Step5: Simplify fifth equation

$3(n+1)=3n+1$
$3n+3=3n+1$
$3=1$ (false)

Step6: Simplify sixth equation

$2n = n$
$n=0$ (one solution)

Step7: Simplify seventh equation

$2t+6=2(t+3)$
$2t+6=2t+6$
$0=0$ (always true)

Step8: Simplify eighth equation

$7-r=r-7$
$14=2r$
$r=7$ (one solution)

Answer:

No Solution (True for No Values)

• $v+2 = v-2$
• $\frac{1}{2}+x = \frac{1}{3}+x$
• $3(n+1)=3n+1$

One Solution (True for One Value)

• $2n = n$
• $7-r=r-7$

Infinitely Many Solutions (True for All Values)

• $y \cdot (-6) \cdot (-3) = 2 \cdot y \cdot 9$
• $2n = 2n$
• $2t+6=2(t+3)$