Consider the following equation:
[tex]$12 w-27=-34+14 w$[/tex]
Select the equation that has the same solution as the given equation above.
A. [tex]$-2 w+21=-8 w+36$[/tex]
B. [tex]$\frac{9}{7} w=\frac{13}{14} w+2$[/tex]
C. [tex]$-\frac{5}{6} w+2=\frac{13}{3}-\frac{1}{6} w$[/tex]
D. [tex]$-47-10 w=-9 w-40-3 w$[/tex]
Answer (2)
Solve the given equation 12 w − 27 = − 34 + 14 w and find w = 2 7 .
Solve option A, − 2 w + 21 = − 8 w + 36 , and find w = 2 5 .
Solve option B, 7 9 w = 14 13 w + 2 , and find w = 5 28 .
Solve option C, − 6 5 w + 2 = 3 13 − 6 1 w , and find w = − 2 7 .
Solve option D, − 47 − 10 w = − 9 w − 40 − 3 w , and find w = 2 7 .
The equation with the same solution is D .
Explanation
Solving the given equation First, let's solve the given equation for w :
12 w − 27 = − 34 + 14 w
Isolating the term with w Subtract 12 w from both sides: − 27 = − 34 + 2 w
Further isolating w Add 34 to both sides: 7 = 2 w
Solution for w Divide both sides by 2 :
w = 2 7 So, the solution to the given equation is w = 2 7 = 3.5 .
Solving the options Now, let's solve each of the options for w and see which one matches our solution.
Solving option A Option A: − 2 w + 21 = − 8 w + 36 Add 8 w to both sides: 6 w + 21 = 36 Subtract 21 from both sides: 6 w = 15 Divide by 6 :
w = 6 15 = 2 5 = 2.5 This does not match.
Solving option B Option B: 7 9 w = 14 13 w + 2 Multiply both sides by 14 to eliminate fractions: 18 w = 13 w + 28 Subtract 13 w from both sides: 5 w = 28 Divide by 5 :
w = 5 28 = 5.6 This does not match.
Solving option C Option C: − 6 5 w + 2 = 3 13 − 6 1 w Multiply both sides by 6 to eliminate fractions: − 5 w + 12 = 26 − w Add 5 w to both sides: 12 = 26 + 4 w Subtract 26 from both sides: − 14 = 4 w Divide by 4 :
w = 4 − 14 = − 2 7 = − 3.5 This does not match.
Solving option D Option D: − 47 − 10 w = − 9 w − 40 − 3 w Combine like terms on the right side: − 47 − 10 w = − 12 w − 40 Add 12 w to both sides: − 47 + 2 w = − 40 Add 47 to both sides: 2 w = 7 Divide by 2 :
w = 2 7 = 3.5 This matches the solution of the given equation.
Final Answer Therefore, the equation that has the same solution as the given equation is option D.
Examples
When balancing chemical equations, you often need to find equivalent expressions to ensure the number of atoms is the same on both sides. Similarly, in economics, you might manipulate equations representing supply and demand to find equilibrium points. This involves solving linear equations and identifying equivalent forms, just like in our problem. Understanding how to manipulate and solve equations is crucial in many fields for making accurate predictions and decisions.
The original equation is solved to find w = 2 7 . When each option is evaluated, only Option D also yields w = 2 7 . Therefore, the correct answer is Option D.
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