Simplify. (2m3n2)5 10m8n7 32m8n7 32m15n10 10m15n10
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The number of solutions possible to a system of equations that include a conic section depends on the type of equations involved
<h3>How to determine the number of solutions?</h3>A system of equation that has a conic section may or may not have solutions.
It would have a solution if the equations intersect, when plotted on a graph and it would not, if otherwise
A conic section can be any of:
- Parabola
- Ellipse
- Hyperbola
- Circle
<u>Example 1: No solution</u>
Consider the following system of equations
- y = x² - 10 --- parabola equation
- (x - 1)² + (y - 2)² = 5 --- circle
The above system of equations has no solution because they do not intersect when plotted on a graph (see graph 1)
<u>Example 2: One solution</u>
Consider the following system of equations
- x = 2 --- linear equation
- y = x² - 10 --- parabola
The above system of equations has one solution because they intersect at one point (see graph 2)
<u>Example 3: One solution</u>
Consider the following system of equations
- y = 2x + 1 --- linear equation
- y = x² - 10 --- circle
The above system of equations has two solutions because they intersect at two points (see graph 3)
The above examples imply that a system of equations that involves conic section can have as many solutions as possible.
The number of solutions depends on the type of equations involved
The attached graphs illustrate how a system of equations can have 0 to 4 solutions
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