Tamara, Payton, Krish, and Haruki are analyzing the solution sets to the following linear equations. Tamara says that Equation A and Equation C have the same solution. Payton says Equation B and Equation C have the same solution. Krish says all three have the same solution. Haruki says that none of the equations have the same solution. Identify who is correct about the linear equations.
Equation A: 3(4−5h)−7=−12+2h
Equation B: 2(p−3)−1=6p+142
Equation C: 23v−3+83=−1
(1 point)
Responses
Haruki
Krish
Payton
Tamara
To determine who is correct about the linear equations, we need to find the solutions to each equation and then compare them.
Equation A: 3(4−5h)−7=−12+2h
Simplify:
12 - 15h - 7 = -12 + 2h
Combine like terms:
-15h + 5h = -12 - 7 - 12
-10h = -31
Divide by -10:
h = 31/10
Equation B: 2(p−3)−1=6p+142
Simplify:
2p - 6 - 1 = 6p + 142
Combine like terms:
2p - 7 = 6p + 142
Subtract 2p:
-7 = 4p + 142
Subtract 142:
-149 = 4p
Divide by 4:
p = -149/4
Equation C: 23v−3+83=−1
Simplify:
23v + 80 = -1
Subtract 80:
23v = -81
Divide by 23:
v = -81/23
Comparing the solutions:
Tamara is incorrect because the solution to Equation A is h = 31/10, while Equation C has v = -81/23.
Payton is incorrect because the solution to Equation B is p = -149/4, while Equation C has v = -81/23.
Krish is incorrect because the solutions to all three equations are different.
Haruki is correct because none of the equations have the same solution.
Therefore, the correct answer is Haruki.
To determine who is correct about the linear equations, let's solve each equation step by step.
Equation A:
3(4−5h)−7=−12+2h
Distribute the 3:
12 - 15h - 7 = -12 + 2h
Combine the like terms:
5 - 15h = -12 + 2h
Move the variables to one side and the constants to the other side:
-15h - 2h = -12 - 5
-17h = -17
Divide both sides by -17:
h = 1
So, the solution for Equation A is h = 1.
Equation B:
2(p−3)−1=6p+142
Distribute the 2:
2p - 6 - 1 = 6p + 142
Combine the like terms:
2p - 7 = 6p + 142
Move the variables to one side and the constants to the other side:
2p - 6p = 142 + 7
-4p = 149
Divide both sides by -4:
p = -149/4
So, the solution for Equation B is p = -149/4 or -37.25.
Equation C:
23v−3+83=−1
Combine the constants:
23v + 80 = -1
Move the constants to the other side:
23v = -81
Divide both sides by 23:
v = -81/23
So, the solution for Equation C is v = -81/23 or approximately -3.522.
Now let's compare the solutions:
The solution for Equation A is h = 1.
The solution for Equation B is p = -149/4 or -37.25.
The solution for Equation C is v = -81/23 or approximately -3.522.
Based on the solutions, Tamara is correct. Equation A and Equation C have the same solution, which is h = 1.