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Systems of Equations (Comparison of Methods)
Graphing, Substitution, and Elimination Side by Side
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A system of equations is a set of two or more equations that share the same variables. Solving the system means finding values that make all equations true at the same time. Students study systems because they model real situations with multiple conditions, such as cost and quantity or distance and time. Comparing methods helps you choose the fastest and clearest strategy for a given problem.
For two linear equations, the solution is the point where the lines intersect on a graph. You can find that point by graphing, by substitution, or by elimination. Each method uses the same algebra but organizes the work differently. Understanding how the methods connect makes it easier to check answers and recognize when a system has one solution, no solution, or infinitely many solutions.
Key Facts
- A solution to a system is an ordered pair (x, y) that satisfies both equations.
- Example system: y = x + 1 and x + y = 5
- Substitution for the example: x + (x + 1) = 5
- Elimination often starts by writing both equations in standard form: -x + y = 1 and x + y = 5
- For the example, adding the equations gives 2y = 6, so y = 3 and x = 2
- Linear systems can have one solution, no solution, or infinitely many solutions depending on whether the lines intersect, are parallel, or are the same line.
Vocabulary
- System of equations
- A set of equations that use the same variables and are solved together.
- Solution
- The values of the variables that make every equation in the system true.
- Substitution
- A method where one variable expression is replaced into another equation.
- Elimination
- A method where equations are added or subtracted to remove one variable.
- Intersection
- The point where two graphs cross, representing the solution of a linear system.
Common Mistakes to Avoid
- Using a point that works in only one equation, which is wrong because a system solution must satisfy both equations at the same time.
- Making sign errors during elimination, which is wrong because adding or subtracting equations with incorrect signs changes the system and gives a false answer.
- Substituting into the wrong part of an equation, which is wrong because you must replace the entire variable with its equal expression, including parentheses when needed.
- Reading the graph imprecisely, which is wrong because an approximate intersection can lead to a wrong ordered pair if the scale is not checked carefully.
Practice Questions
- 1 Solve the system by substitution: y = 2x - 3 and x + y = 9.
- 2 Solve the system by elimination: 2x + y = 7 and x - y = 2.
- 3 A system has equations y = 3x + 2 and y = 3x - 4. Without solving by algebra, explain how you know how many solutions the system has.