Select the correct location on the table.
Three students solved linear 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: [tex]-\frac{1}{3} x+2=\frac{5}{2}[/tex]
Isabelle: [tex]12 x+25=7[/tex]
Steven: ???
Answer (1)
Solve Tristan's equation: − 3 1 x + 2 = 2 5 to find x = − 2 3 .
Solve Isabelle's equation: 12 x + 25 = 7 to find x = − 2 3 .
Solve each of Steven's possible equations and compare the solutions to − 2 3 .
Identify that 6 x − 5 = − 14 has the solution x = − 2 3 .
The correct equation is 6 x − 5 = − 14 .
Explanation
Finding the Common Solution First, we need to solve Tristan's and Isabelle's equations to find their common solution. This will allow us to determine which of Steven's possible equations has the same solution.
Solving Tristan's Equation Tristan's equation is − 3 1 x + 2 = 2 5 . To solve for x , we first subtract 2 from both sides: − 3 1 x = 2 5 − 2 = 2 5 − 2 4 = 2 1 Then, we multiply both sides by -3: x = 2 1 × ( − 3 ) = − 2 3 So, Tristan's solution is x = − 2 3 .
Solving Isabelle's Equation Isabelle's equation is 12 x + 25 = 7 . To solve for x , we first subtract 25 from both sides: 12 x = 7 − 25 = − 18 Then, we divide both sides by 12: x = 12 − 18 = − 2 3 So, Isabelle's solution is x = − 2 3 .
Listing Steven's Possible Equations Since both Tristan and Isabelle have the same solution, x = − 2 3 , we need to find which of Steven's possible equations also has this solution. We will solve each of Steven's equations for x and check if the solution is x = − 2 3 .
The possible equations for Steven are:
− 6 5 x + 3 2 = − 12 7
4 x + 9 = 15
− x + 3 4 = 2
− 18 x + 5 = 17
2 1 x + 5 = 3 14
6 x − 5 = − 14
Solving Steven's Equations Now, let's solve each of Steven's possible equations:
− 6 5 x + 3 2 = − 12 7 − 6 5 x = − 12 7 − 3 2 = − 12 7 − 12 8 = − 12 15 = − 4 5 x = − 4 5 × − 5 6 = 20 30 = 2 3
4 x + 9 = 154 x = 15 − 9 = 6 x = 4 6 = 2 3
− x + 3 4 = 2 − x = 2 − 3 4 = 3 6 − 3 4 = 3 2 x = − 3 2
− 18 x + 5 = 17 − 18 x = 17 − 5 = 12 x = − 18 12 = − 3 2
2 1 x + 5 = 3 14 2 1 x = 3 14 − 5 = 3 14 − 3 15 = − 3 1 x = − 3 1 × 2 = − 3 2
6 x − 5 = − 146 x = − 14 + 5 = − 9 x = 6 − 9 = − 2 3
Identifying Steven's Equation Comparing the solutions, we see that only equation 6, 6 x − 5 = − 14 , has the same solution as Tristan and Isabelle, which is x = − 2 3 .
Final Answer Therefore, the correct equation that Steven solved is 6 x − 5 = − 14 .
Location on the Table The equation that Steven solved is 6 x − 5 = − 14 , which is located in the bottom right cell of the table.
Examples
Imagine you and your friends are trying to figure out who solved a math problem correctly. You each have different equations, but you all got the same answer. By solving your equations and comparing the results, you can determine which equation leads to the correct solution. This is similar to how engineers might check different designs or models to ensure they all meet the same performance criteria, ensuring accuracy and reliability in their work.
