Question

number of solutions | equation
no solution | $2(r + 4) = 2(r + 3)$
exactly one solution | $6(p + 3) + 5 = 3 + 5(2p + 4) - 4p$
infinitely many solutions | $4(m + 3) + 2 = 3(m + 6)$
how many solutions equation have?
drag each equation category in the table the reset button to s
take another look be careful when simplifying the equations. were you able to simplify the equation to a single solution, like $x = 1$? if not, the equation does not have exactly one solution.

Explanation:

Step1: Simplify first equation

Expand both sides:
$2(r+4)=2(r+3)$
$2r+8=2r+6$
Subtract $2r$ from both sides:
$8=6$

Step2: Simplify second equation

Expand and combine like terms:
$6(p+3)+5=3+5(2p+4)-4p$
$6p+18+5=3+10p+20-4p$
$6p+23=6p+23$
Subtract $6p+23$ from both sides:
$0=0$

Step3: Simplify third equation

Expand and solve for $m$:
$4(m+3)+2=3(m+6)$
$4m+12+2=3m+18$
$4m+14=3m+18$
Subtract $3m$ from both sides:
$m+14=18$
Subtract 14 from both sides:
$m=4$

Step4: Match to solution categories

• Contradiction ($8=6$): No solution
• Identity ($0=0$): Infinitely many solutions
• Single value ($m=4$): Exactly one solution

Answer:

Number of Solutions	Equation
Exactly one solution	$2(r+4)=2(r+3)$
Infinitely many solutions	$6(p+3)+5=3+5(2p+4)-4p$

Corrected matching:

• No solution: $2(r+4)=2(r+3)$
• Exactly one solution: $4(m+3)+2=3(m+6)$
• Infinitely many solutions: $6(p+3)+5=3+5(2p+4)-4p$