Question
Emma was given a system of equations to solve by graphing. Which statement correctly identifies Emma’s error?
Emma’s Graph
On a coordinate plane, a line with equation y = negative one-third x + 2 goes through (0, 1) and (3, 0). A line with equation y = 2 x minus 5 goes through (0, negative 5) and (3, 1).
Line 1 should have a y-intercept at (0, 2).
Line 2 should have a y-intercept at (0, 2).
Line 1 should have a slope of 2.
Line 2 should have a slope of –5.
Emma’s Graph
On a coordinate plane, a line with equation y = negative one-third x + 2 goes through (0, 1) and (3, 0). A line with equation y = 2 x minus 5 goes through (0, negative 5) and (3, 1).
Line 1 should have a y-intercept at (0, 2).
Line 2 should have a y-intercept at (0, 2).
Line 1 should have a slope of 2.
Line 2 should have a slope of –5.
Answers
GPT-4o mini
Based on the equations provided and Emma's graph, we can analyze the statements regarding her errors:
1. **Line 1 (y = -1/3 x + 2)** has a y-intercept at (0, 2), so this statement about Line 1 having the y-intercept at (0, 2) is correct.
2. **Line 2 (y = 2x - 5)** has a y-intercept at (0, -5), not (0, 2), so this statement about Line 2 having the y-intercept at (0, 2) is incorrect.
3. **Line 1** has a slope of -1/3, not 2, so this statement about Line 1 having a slope of 2 is incorrect.
4. **Line 2** has a slope of 2 (as indicated in the equation y = 2x - 5), not -5, so this statement about Line 2 having a slope of -5 is also incorrect.
Thus, the statements that correctly identify Emma's errors are:
- "Line 2 should have a y-intercept at (0, 2)."
- "Line 1 should have a slope of 2."
- "Line 2 should have a slope of -5."
However, the most accurate error relating to the graph is that the slope and y-intercept of each line were misrepresented based on their given equations. The primary errors seem to stem from the incorrect slopes and y-intercepts on the graph compared to the algebraic equations.
Overall, a predominant correct identification of Emma's main error is that **Line 1 should have a slope of -1/3 and should not incorrectly be represented with a slope of 2.**
1. **Line 1 (y = -1/3 x + 2)** has a y-intercept at (0, 2), so this statement about Line 1 having the y-intercept at (0, 2) is correct.
2. **Line 2 (y = 2x - 5)** has a y-intercept at (0, -5), not (0, 2), so this statement about Line 2 having the y-intercept at (0, 2) is incorrect.
3. **Line 1** has a slope of -1/3, not 2, so this statement about Line 1 having a slope of 2 is incorrect.
4. **Line 2** has a slope of 2 (as indicated in the equation y = 2x - 5), not -5, so this statement about Line 2 having a slope of -5 is also incorrect.
Thus, the statements that correctly identify Emma's errors are:
- "Line 2 should have a y-intercept at (0, 2)."
- "Line 1 should have a slope of 2."
- "Line 2 should have a slope of -5."
However, the most accurate error relating to the graph is that the slope and y-intercept of each line were misrepresented based on their given equations. The primary errors seem to stem from the incorrect slopes and y-intercepts on the graph compared to the algebraic equations.
Overall, a predominant correct identification of Emma's main error is that **Line 1 should have a slope of -1/3 and should not incorrectly be represented with a slope of 2.**