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What is the solution to a linear system represented by parallel lines?
Mathematics
Grade 8 (Junior High School)
Question Content
If the parallel lines graphed below represent a linear system of equations, what can be said about the system of equations?\nA. There is no solution.\nB. There is only one solution.\nC. There are only two solutions.\nD. There are infinite solutions.
Correct Answer
A
Detailed Solution Steps
1
Step 1: Recall the definition of a solution to a system of linear equations graphed on a coordinate plane: a solution is a point (x,y) that lies on both lines, meaning it is an intersection point of the two lines.
2
Step 2: Analyze the property of parallel lines: Parallel lines are defined as lines that never intersect, no matter how far they are extended in either direction.
3
Step 3: Connect the two concepts: Since the parallel lines in the graph never intersect, there is no point that lies on both lines. Therefore, the system of linear equations has no solution.
Knowledge Points Involved
1
Solutions to Systems of Linear Equations (Graphical Interpretation)
When solving a system of linear equations by graphing, the solution(s) correspond to the intersection points of the lines. If lines intersect once, there is 1 unique solution; if lines are the same (coinciding), there are infinite solutions; if lines are parallel, there are no solutions.
2
Definition of Parallel Lines
In a coordinate plane, parallel lines are lines that have identical slopes but different y-intercepts. They maintain a constant distance between each other and never intersect, even when extended infinitely.
3
Consistent and Inconsistent Systems
A consistent system of linear equations has at least one solution (either one or infinite). An inconsistent system has no solutions, which occurs exactly when the system is represented by parallel, non-coinciding lines.
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