Examining equations and parameters
I’m trying to analyze the equations in the prompt: “y = 5?x - 4 and y = -ax + 3a.” It looks like it might be “y = 5ax - 4.” I wonder if it's from an algebra textbook, specifically exercises 110-113 where questions about intersection, parallel lines, and coincident lines are presented. I'll review the exercises to clarify the parameters defining these linear functions, including the equations presented for question 111.Examining equations
I'm working through the equations for this math problem. For one of the equations, I found \(a = \frac{5}{3}\) but realized the differences in intercepts indicate parallel lines. The other equation provided \(a = 1\), but that results in coinciding lines rather than distinct parallel graphs. I need to consider how concise the user wants the answer. The Russian math terminology might be key here, so I'll keep that in mind.Considering parallel lines
I'm thinking about the phrase "графики функций параллельны" and whether it might imply some coincidence. I need to check standard textbooks on the topic. The definition states that two lines are parallel if they don't intersect. Coincident lines might be seen as parallel in some advanced contexts, but usually, schools exclude them because they share infinitely many points. So, I'd like to clarify this with further exercises.
111:
в) \(5a\ne -a\Rightarrow a\ne0\).
г) Нет таких значений: при \(a=0\) графики совпадают, при \(a\ne0\) — параллельны.
112:
в) \(a-2=3-2a\Rightarrow a=\frac53\).
г) Нет таких значений: при \(a=1\) графики совпадают \((y=-x+5)\).