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Three students solved their equations and compared the solutions. They found that all three of the equations had the same solution.



The equations for two of the students, Tristan and Isabelle, are shown below. Determine which is a possible equation that the third student, Steven, solved.



Tristan: $-\frac{1}{3} x+2=\frac{5}{2}$

Isabelle: $12 x+25=7$

Steven: ???



$-\frac{5}{6} x+\frac{2}{3}=-\frac{7}{12}$		$4 x+9=15$$-x+\frac{4}{3}=2$

$-18 x+5=17$		$\frac{1}{2} x+5=\frac{14}{3}$$6 x-5=-14$

Verified Answer :

  • First, solve Tristan's equation \(-\frac{1}{3}x + 2 = \frac{5}{2}\) to find the common solution \(x = -1.5\).
  • Second, solve Isabelle's equation \(12x + 25 = 7\) to verify the common solution \(x = -1.5\).
  • Finally, evaluate each of the six candidate equations and find that \(6x - 5 = -14\) results in \(x = -1.5\).
  • The correct equation is \(\boxed{6x - 5 = -14}\).

Explanation

  1. Solve Tristan and Isabelle's equations

We begin by determining the common solution \(x\) for the two given equations from Tristan and Isabelle.
Tristan's equation is:
\[-\frac{1}{3}x + 2 = \frac{5}{2}\]
Subtracting 2 from both sides:
\[-\frac{1}{3}x = \frac{5}{2} - 2 = \frac{5}{2} - \frac{4}{2} = \frac{1}{2}\]
Multiplying both sides by -3:
\[x = \frac{1}{2} \times (-3) = -\frac{3}{2} = -1.5\]

Isabelle's equation is:
\[12x + 25 = 7\]
Subtracting 25 from both sides:
\[12x = 7 - 25 = -18\]
Dividing by 12:
\[x = -\frac{18}{12} = -\frac{3}{2} = -1.5\]
Both equations yield the same solution: \(x = -1.5\).

  1. Identify Steven's equation

Now we evaluate the six candidate equations to find which one results in \(x = -1.5\):

  1. \(-\frac{5}{6}x + \frac{2}{3} = -\frac{7}{12}\):

\[-\frac{5}{6}x = -\frac{7}{12} - \frac{8}{12} = -\frac{15}{12} \implies x = (-\frac{15}{12}) \times (-\frac{6}{5}) = 1.5\]

  1. \(4x + 9 = 15\):

\[4x = 6 \implies x = \frac{6}{4} = 1.5\]

  1. \(-x + \frac{4}{3} = 2\):

\[-x = 2 - \frac{4}{3} = \frac{2}{3} \implies x = -\frac{2}{3} \approx -0.67\]

  1. \(-18x + 5 = 17\):

\[-18x = 12 \implies x = -\frac{12}{18} = -\frac{2}{3} \approx -0.67\]

  1. \(\frac{1}{2}x + 5 = \frac{14}{3}\):

\[\frac{1}{2}x = \frac{14}{3} - \frac{15}{3} = -\frac{1}{3} \implies x = -2/3 \approx -0.67\]

  1. \(6x - 5 = -14\):

\[6x = -9 \implies x = -\frac{9}{6} = -\frac{3}{2} = -1.5\]
The only equation that yields the solution \(x = -1.5\) is the sixth one.

  1. Final conclusion

Based on our step-by-step analysis, the equation from the provided list that shares the same solution as Tristan's and Isabelle's is \(6x - 5 = -14\). The common solution for all three students is \(-1.5\). Therefore, the correct equation for Steven is \(\boxed{6x - 5 = -14}\).

Examples

Solving linear equations is a fundamental skill used in many practical scenarios, such as budgeting. If you are planning a project where costs are split among participants, you might set up an algebraic equation to ensure everyone pays their fair share. For instance, if you have a base cost and a variable cost dependent on the number of hours worked, solving for the number of hours \(x\) allows you to verify that the budget is balanced correctly. Mastering this skill ensures accuracy in managing finances or resources.

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