Exercises
Build confidence with the fundamentals of systems of linear equations. This introductory quiz covers solving two-variable systems, recognizing systems with no solution, choosing useful operations such as elimination, and interpreting solutions from graphs. Explore how intersecting lines relate to consistent systems, why homogeneous systems always have at least the trivial solution, and what identical systems reveal about solution sets. Ideal for students learning algebraic methods for solving and analyzing linear equations.
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To solve the system, substitute x = y + 1 into 2x + 3y = 6, resulting in 2(y+1) + 3y = 6 which simplifies to 5y + 2 = 6 or y = 4/5. Then x = 9/5. Solution: (9/5, 4/5).
A system with parallel lines has no solutions as they never intersect. The system x - y = 2 and x - y = 5 represents parallel lines.
The substitution method involves solving one equation for one variable and substituting this into the other equation, which is effective for systems with different coefficients.
A consistent system represented by the intersection of two lines has exactly one solution, which is the point where both lines intersect.
A homogeneous system always has at least one solution, the trivial solution where all variables equal zero. It has infinite solutions if the determinant is zero.
If two systems of equations are identical, they consist of the same line repeated, leading to infinitely many solutions as every point on the line is a solution.
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