Since the two linear relations are parallel to each other, no solution to the system exists. Hence, Samia's conclusion is correct.
How many solutions do a system of linear lines have?
A system of equations, involving two linear lines has solutions as follows:
- Unique Solution: When the lines intersect, the point of intersection is the solution.
- No Solution: When the lines are parallel, no solution exists.
- Infinite Solution: When the lines coincide, then all points on the line become solutions, giving an infinite number of solutions.
How to solve the question?
In the question, we are informed that Samia created the given tables for a linear system, and concluded that no solution exists for the system.
We are asked to comment on her conclusion.
To check for her conclusion, we calculate the slopes of both the lines to check whether the relations are parallel or not, as, for parallel relations, the slopes are equal.
Slope can be calculated using the formula, m = (y₂ - y₁)/(x₂ - x₁), when (x₁, y₁), and (x₂, y₂) are the points on the line.
Thus, the slope for:-
Relation 1, is m₁ = (22 - 8)/(4 - (-3)) = 14/7 = 2.
Relation 2, is m₂ = (12 - (-2))/(4 - (-3)) = 14/7 = 2.
Since the slope of the two relations is equal, that is, m₁ = m₂, and they are not coinciding with each other, we can say that the two relations are parallel to each other.
Since the two linear relations are parallel to each other, no solution to the system exists. Hence, Samia's conclusion is correct.
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