Question
Which type(s) of equation can have infinite solutions? \( \square \) linear-linear linear-quadratic quadratic-quadratic DONE
Ask by King May. in the United States
Feb 26,2025
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Answer
Equations that can have infinitely many solutions are:
- Linear-linear
- Quadratic-quadratic
Solution
A linear–linear equation can have infinitely many solutions if the two linear expressions represent the same line (i.e., they are identical after simplification).
Similarly, a quadratic–quadratic equation can have infinitely many solutions if the two quadratics are identical, meaning every term and coefficient matches when simplified.
However, a linear–quadratic equation (one side linear and the other quadratic) cannot have infinitely many solutions because a nondegenerate quadratic is a parabola and a line cannot coincide with a parabola over all x‐values. (If the quadratic were degenerated to a linear expression, then it wouldn’t be a true quadratic–linear case anymore.)
So, the types of equations that can have infinitely many solutions are:
– Linear–linear
– Quadratic–quadratic
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Simplify this solution The Deep Dive
When it comes to infinite solutions, linear-linear equations are prime contenders! These are typically represented as two lines that overlap completely, meaning every point on one line is also on the other. Imagine two identical highways running parallel forever; they are, essentially, the same path! On the other hand, quadratic-quadratic equations can also produce infinite solutions under specific conditions, such as when both parabolas are the same or completely overlap in certain sections. It’s like two magical roller coasters that perfectly align throughout their entire ride! Such scenarios create a fun mathematical adventure with solutions galore!
