Question
Sylvie solved this system of equations.
y = 5x + 2,
y = 8 – x
8 – x = 5x + 2
8 = 6x + 2
6 = 6x
x = 1
8 – 1 = 5(1) + 2
7 = 7
There are infinite solutions to this system.
When Sylvie verified the solution on a graph, the lines intersected at one point. What was her error?
She forgot to put both equations in slope-intercept form.
She made an arithmetic mistake in the last step.
She found the incorrect value of x.
She did not substitute the value of x into one of the original equations to find y.
y = 5x + 2,
y = 8 – x
8 – x = 5x + 2
8 = 6x + 2
6 = 6x
x = 1
8 – 1 = 5(1) + 2
7 = 7
There are infinite solutions to this system.
When Sylvie verified the solution on a graph, the lines intersected at one point. What was her error?
She forgot to put both equations in slope-intercept form.
She made an arithmetic mistake in the last step.
She found the incorrect value of x.
She did not substitute the value of x into one of the original equations to find y.
Answers
GPT-4o mini
Sylvie's error lies in the conclusion she drew about the number of solutions to the system of equations. The statements made in each step leading to the conclusion were correct, but her final assertion about "infinite solutions" was incorrect, because the graph of the two equations shows that they intersect at exactly one point.
Here's a breakdown of why she is incorrect:
1. She correctly set the two equations equal to each other and solved for \( x \).
2. She found \( x = 1 \) correctly.
3. When substituting \( x = 1 \) back into the equations, both equations yield \( y = 7 \), confirming that the point (1, 7) is the only solution.
4. Since the two equations represent distinct lines with different slopes (5 and -1), they can only intersect at one point.
So, the most accurate summary of her error is that she concluded there were "infinite solutions" instead of recognizing that there is one unique solution. Therefore, the correct answer is:
**She found the incorrect value of x.** (Although this statement isn't the best representation of her error, the other options do not directly address the issue). But in essence, the more precise answer is: **She incorrectly concluded there were infinite solutions to the system.**
Here's a breakdown of why she is incorrect:
1. She correctly set the two equations equal to each other and solved for \( x \).
2. She found \( x = 1 \) correctly.
3. When substituting \( x = 1 \) back into the equations, both equations yield \( y = 7 \), confirming that the point (1, 7) is the only solution.
4. Since the two equations represent distinct lines with different slopes (5 and -1), they can only intersect at one point.
So, the most accurate summary of her error is that she concluded there were "infinite solutions" instead of recognizing that there is one unique solution. Therefore, the correct answer is:
**She found the incorrect value of x.** (Although this statement isn't the best representation of her error, the other options do not directly address the issue). But in essence, the more precise answer is: **She incorrectly concluded there were infinite solutions to the system.**