Solve the following system of equations. What is the correct solution, and what error might have been made? 6x - 2y = -6 11 = y - 5x
Understand the Problem
The problem states that Tim incorrectly solved the system of equations and got x = -9, y = -4. We need to find the correct solution to the system of equations, and identify the error Tim made when solving the system.
Answer
a. $x = -4$, $y = -9$ b. Error in the substitution or simplification process.
Answer for screen readers
a. The correct solution is $x = -4$, $y = -9$. b. Tim may have made an error in the substitution or simplification process, possibly switching the variables x and y at some point.
Steps to Solve
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Rewrite the second equation Rewrite the second equation in the form $y = ...$ $11 = y - 5x$ $y = 5x + 11$
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Substitute the second equation into the first equation Substitute $y = 5x + 11$ into the first equation $6x - 2y = -6$ $6x - 2(5x + 11) = -6$
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Solve for x Simplify and solve the equation for $x$: $6x - 10x - 22 = -6$ $-4x = 16$ $x = -4$
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Solve for y Substitute $x = -4$ into the equation $y = 5x + 11$ $y = 5(-4) + 11$ $y = -20 + 11$ $y = -9$
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State the correct solution The correct solution is therefore $x = -4$ and $y = -9$.
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Check Tim's solution in the equations Check if Tim's solution ($x = -9, y = -4$) satisfies either equation: Equation 1: $6x - 2y = -6$ $6(-9) - 2(-4) = -54 + 8 = -46 \neq -6$. So the first equation is not satisfied. Equation 2: $11 = y - 5x$ $11 = -4 - 5(-9) = -4 + 45 = 41 \neq 11$. So the second equation is not satisfied either.
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Identify the potential error A possible error Tim made could be that he substituted incorrectly, or that he made a mistake when solving the system after the substitution. For example, he may have added instead of subtracted when simplifying. Let's plug Tim's solution into the first equation: $6(-9) - 2(-4) = -6$ $-54 + 8 = -6$ $-46 = -6$ If Tim had added 54 to both sides instead of subtracting -8, then he would have -46 + 54 = 8 on the left side and -6 + 54 = 48 on the right side. If he divided 48 by -8 he would have ended up with x = -6. Then he could have thought that 11 = -4 - 5(-9) = -4 + 45. If those were reversed, then 11 = -9 - 5(-4), if that was Tim's thought process, he might have switched x and y after solving which would lead to his answer. This is only one possibility.
a. The correct solution is $x = -4$, $y = -9$. b. Tim may have made an error in the substitution or simplification process, possibly switching the variables x and y at some point.
More Information
Systems of equations can be solved using several methods, including substitution, elimination, and graphing. The solution to a system of equations is the ordered pair (x, y) that satisfies both equations simultaneously.
Tips
A common mistake is making an arithmetic error when simplifying or substituting. Another mistake is not distributing correctly when expanding expressions like $-2(5x + 11)$. It is also possible to mix up the $x$ and $y$ values during substitution or when stating the final answer.
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