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.

Question

Equation A: 3(4−5h)−7=−12+2h

Equation B: 2(p−3)−1=6p+142

Equation C: 23v−3+83=−1

(1 point)

Answer 1

Let's analyze each equation step by step to determine their solution sets.

Equation A:

$$3(4-5h)-7=-12+2h$$ First, we simplify the left side: $$3(4-5h)=12-15h$$ So, $$12-15h-7=-12+2h$$ This simplifies to: $$5-15h=-12+2h$$ Now, combine like terms: $$5+12=15h+2h$$ which results in: $$17=17h$$ Dividing by 17 gives: $$h=1$$

Equation B:

$$2(p-3)-1=6p+142$$ First, we simplify: $$2p-6-1=6p+142$$ This becomes: $$2p-7=6p+142$$ Rearranging the equation: $$-7-142=6p-2p$$ This simplifies to: $$-149=4p$$ Dividing by 4 gives: $$p=-\frac{149}{4}$$

Equation C:

$$23v-3+83=-1$$ Combine like terms: $$23v+80=-1$$ Now, isolate $v$: $$23v=-1-80$$ This simplifies to: $$23v=-81$$ Dividing by 23 gives: $$v=-\frac{81}{23}$$

Summary of Solutions:

• Equation A: $h=1$
• Equation B: $p=-\frac{149}{4}$
• Equation C: $v=-\frac{81}{23}$

Now let's analyze the statements:

• Tamara says A and C have the same solution. (False, $1\ne -\frac{81}{23}$)
• Payton says B and C have the same solution. (False, $-\frac{149}{4}\ne -\frac{81}{23}$)
• Krish says all three have the same solution. (False)
• Haruki says that none of the equations have the same solution. (True)

Conclusion:

Haruki is correct. None of the equations have the same solution.