Systems of Equations and Inequalities
Grade 10 · Math · Study Guide
Solving and graphing systems of linear equations and inequalities in two variables.
A system of equations is a set of two or more equations that share the same variables. A solution to the system is an ordered pair (x, y) that satisfies all equations simultaneously — in other words, it's the point where the graphs intersect.
There are three main algebraic methods to solve a linear system. Graphing works by plotting both lines and identifying the intersection point, but it's imprecise for non-integer answers. Substitution involves solving one equation for a variable and substituting that expression into the other equation. Elimination (also called addition method) adds or subtracts the equations to cancel out one variable, then solves for the other.
Systems can have one solution (lines intersect at one point), no solution (parallel lines that never meet), or infinitely many solutions (the same line written two different ways). If you end up with a false statement like 0 = 5, there's no solution. A true statement like 0 = 0 means infinitely many solutions.
Systems of inequalities involve graphing each inequality as a shaded half-plane and identifying the overlapping region — called the feasible region. This concept is the foundation of linear programming, used by businesses to maximize profit or minimize cost given real-world constraints like budget limits and production capacity.
| System Type | Number of Solutions | Graph Description |
|---|---|---|
| Consistent & Independent | Exactly 1 | Lines intersect at one point |
| Inconsistent | None | Lines are parallel (never intersect) |
| Consistent & Dependent | Infinitely many | Lines are identical (same line) |
Key terms
- System of Equations
- Two or more equations with the same variables solved simultaneously.
- Solution of a System
- An ordered pair (x, y) that satisfies all equations in the system.
- Substitution Method
- Solving one equation for a variable and plugging that expression into the other equation.
- Elimination Method
- Adding or subtracting equations to eliminate one variable.
- Feasible Region
- The overlapping shaded area that satisfies all inequalities in a system.
Flashcards
- System of equations
- Two or more equations sharing the same variables, solved at the same time
- Solution to a system
- The ordered pair (x,y) that satisfies all equations in the system
- Substitution method
- Solve one equation for a variable, substitute into the other
- Elimination method
- Add or subtract equations to cancel one variable
- Inconsistent system
- No solution; the lines are parallel
- Dependent system
- Infinitely many solutions; the equations describe the same line
Practice questions
What does the solution to a system of two linear equations represent graphically?
- The slope of both lines
- The point of intersection
- The y-intercepts
- The area between the lines
Show answer
Answer: The point of intersection
The solution to a system is the point where both lines cross — their intersection.
A system of two linear equations can have exactly two solutions.
Show answer
Answer: False
Two lines can intersect at 0, 1, or infinitely many points — never exactly 2.
When solving by elimination, you add or subtract equations to ___ one variable.
Show answer
Answer: eliminate
The goal of elimination is to make one variable disappear by adding or subtracting the equations.
Solving the system y = 2x + 1 and y = 2x - 3, you get:
- One solution
- No solution
- Infinitely many solutions
- Two solutions
Show answer
Answer: No solution
Both lines have slope 2 but different y-intercepts (1 vs -3), so they are parallel and never intersect — no solution.
The feasible region in a system of inequalities is where all inequalities are satisfied simultaneously.
Show answer
Answer: True
The feasible region is the intersection of all individual inequality solution regions.
Fun facts
- Linear programming — optimizing within a system of inequalities — was developed during World War II to figure out the most efficient way to supply Allied troops.
- Every GPS calculation involves solving a system of equations from multiple satellites simultaneously to pinpoint your location in three dimensions.
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