Systems of Linear Equations
A system asks for values that satisfy several conditions at the same time. In this unit, you will solve systems graphically and algebraically, organize them with matrices, describe regions that satisfy multiple inequalities, and use those regions to make optimized decisions.
Lesson Map
Each lesson matches one subsection of Chapter 2 in the W.A. Algebra 2 text. The methods change, but the central idea does not: a solution to a system must make every equation or inequality true at the same time.
| Lesson | Textbook subsection | Focus | You should be able to… |
|---|---|---|---|
| Lesson 1 | 2.1 | Graphing & substitution | Find the intersection of two lines by graphing or substitution and classify systems as independent, dependent, or inconsistent. |
| Lesson 2 | 2.2 | Elimination | Create opposite coefficients, eliminate one variable, back-substitute, and recognize special cases. |
| Lesson 3 | 2.3 | Matrices | Translate a system into an augmented matrix, use row operations to solve it, and interpret reduced row-echelon form. |
| Lesson 4 | 2.4 | Systems of inequalities & feasibility regions | Graph several inequalities, identify their common feasible region, locate vertices, and test an objective function. |
| Lesson 5 | 2.5 | Applications & linear programming | Turn real constraints into inequalities, construct an objective function, and determine a maximum or minimum. |
| Performance Task | Unit synthesis | Motion intersection & engineering constraints | Measure two moving objects, predict their meeting point with a system, verify experimentally, and optimize a constrained setup. |
Content Summary
Each lesson begins with the meaning of the method, then shows the algebra students should be able to reproduce independently.
Solving Systems by Graphing and Substitution
A system of equations is a set of equations considered together. For two linear equations, the solution is the ordered pair (x, y) that makes both equations true. Graphically, that is the point where the two lines intersect.
Graphing
y = −x + 5
Graph both equations on the same plane. If the lines cross once, the intersection is the single solution. Setting the equations equal checks the graph: 2x − 1 = −x + 5, so 3x = 6, x = 2 and y = 3.
Substitution
2x + y = 11
If one equation is already solved for a variable, replace that variable in the other equation: 2x + (3x + 1) = 11 → 5x = 10 → x = 2, and y = 3(2) + 1 = 7. The solution is (2, 7).
One crossing point means one solution. That single point is the only pair of values that satisfies both equations at once.
Three possible system types
Independent
The lines cross once, so there is exactly one solution.
Inconsistent
The lines are parallel and distinct, so there is no solution.
Dependent
The equations describe the same line, so there are infinitely many solutions.
Solving Systems by Elimination
Elimination uses the addition or subtraction properties of equality to make one variable disappear. The best target is usually the variable whose coefficients are already opposites or can become opposites with simple multiplication.
Worked example — coefficients already cancel.
2x − y = 1
Add the equations, then back-substitute into either original equation:
The solution is (2, 3).
Worked example — multiply first. Multiply the first equation by 3 and the second by 2 so the y-coefficients become opposites.
5x − 3y = −1 → 10x − 6y = −2
Then substitute back to find y. Fractional solutions are allowed; do not assume an arithmetic error merely because the intersection is not an integer point.
Solving Systems with Matrices
A matrix stores the coefficients and constants of a system without repeatedly writing the variables. This chapter uses augmented matrices and row operations to transform a system into a form where the solution can be read directly.
Translate the system
x − y = 1
[ 1 −1 | 1 ]
Keep the variables in the same order in every equation. A missing variable needs a coefficient of zero.
Target form
[ 0 1 | b ]
For a two-variable system, this form means x = a and y = b, with no further algebra required.
Allowed row operations
- Swap two rows.
- Multiply a row by any nonzero number.
- Add a multiple of one row to another row.
Worked example. Start with the augmented matrix, replace Row 2 with Row 2 − 2(Row 1), divide Row 2 by 5, then add Row 2 to Row 1.
[ 2 3 | 7 ] → [ 0 5 | 5 ] → [ 0 1 | 1 ]
Therefore (x, y) = (2, 1).
Systems of Linear Inequalities and Feasibility Regions
An equation describes a boundary line. An inequality describes an entire half-plane. When several inequalities are graphed together, the points satisfying all of them form the feasibility region.
Boundary rules
- Use a solid boundary for ≤ or ≥.
- Use a dashed boundary for < or >.
- Shade the side of the line that satisfies the inequality.
- The system solution is the overlap of all shaded regions.
Example constraints
y ≥ 2
x ≥ 1
The feasible region is above y = 2, to the right of x = 1, and below y = −x + 8.
Three constraints cut the plane down to one triangle. Every point inside or on that triangle satisfies all three inequalities; its corners are the vertices to test.
Vertices and objective functions
When a linear objective such as profit or cost is optimized over a polygonal feasible region, the maximum or minimum occurs at a vertex of the region. Find each relevant intersection, substitute each vertex into the objective function, and compare the results.
Using the vertices above:
The maximum is 30 at (6, 2).
Applications of Linear Systems: Linear Programming
Linear programming turns a real decision into mathematics. The process is consistent: define variables, write constraints, graph the feasible region, write a profit or cost objective, find the vertices, evaluate the objective at each viable vertex, and interpret the best result.
Worked example — science kits. A class can build basic kits x and advanced kits y. Each basic kit uses 2 batteries and 1 sensor; each advanced kit uses 1 battery and 3 sensors. There are 24 batteries and 30 sensors available, and the class wants at least 2 of each kit. A basic kit earns 5 points and an advanced kit earns 8 points in a design challenge.
x + 3y ≤ 30
x ≥ 2, y ≥ 2
The important work is not merely graphing. Students must explain what each inequality means, why x and y cannot be negative or fractional when counting physical objects, and why the selected vertex makes sense in the original situation.
Vocabulary
These terms tie the algebra of systems to the graphs and the applications that use them.
Two or more equations considered together.
A value or ordered pair that satisfies every equation in the system simultaneously.
A system with exactly one solution.
A system whose equations represent the same line and therefore have infinitely many solutions.
A system with no solution, such as distinct parallel lines.
Replacing a variable with an equivalent expression from another equation.
Combining equations so that one variable cancels.
A rectangular array of numbers used to organize coefficients, constants, or data.
A matrix that contains both the coefficient columns and the constant column of a system.
An allowed operation used to transform a matrix without changing its solution set.
A simplified matrix form from which variable values can be read directly.
Two or more inequalities whose common solution set is sought.
The set of all points satisfying every constraint in a system of inequalities.
A limitation written as an equation or inequality.
The quantity to be maximized or minimized, such as profit, cost, output, or efficiency.
Optimizing a linear objective subject to linear constraints.
Sample Problems
Fifteen problems that mirror the skills and progression of Chapter 2 rather than copying its exercise set. Work each one on paper first — the solutions are one click away, which makes them very easy to read too early.
Lesson 1 — Graphing & Substitution
Solve a system by graphing
Find the intersection of y = 2x + 1 and y = −x + 7.
Show solution
The lines intersect at (2, 5), so the system is independent.
Solve by substitution
Solve y = 4x − 6 and 3x + y = 8.
Show solution
Solution: (2, 2).
Classify a system
Classify the system y = 3x − 4 and 6x − 2y = 8.
Show solution
The equations are identical, so the system is dependent with infinitely many solutions.
Identify no solution
Solve 2x + y = 5 and 4x + 2y = 14.
Show solution
Doubling the first equation gives 4x + 2y = 10, not 14. The left sides have proportional coefficients but different constants, so the lines are parallel and the system is inconsistent.
Lesson 2 — Elimination
Eliminate immediately
Solve 5x + 2y = 18 and 3x − 2y = 6.
Show solution
Solution: (3, 3/2).
Create opposite coefficients
Solve 2x + 3y = 13 and 5x − 2y = 4.
Show solution
Multiply the first equation by 2 and the second by 3:
15x − 6y = 12
Solution: (2, 3).
Recognize a dependent system through elimination
Solve 3x − 6y = 9 and −x + 2y = −3.
Show solution
Multiply the second equation by 3:
Adding gives 0 = 0. The system is dependent and has infinitely many solutions.
Lesson 3 — Matrices
Write an augmented matrix
Write the augmented matrix for 3x − 2y = 7 and 5x + 4y = 1.
Show solution
[ 5 4 | 1 ]
Read a reduced matrix
Interpret the reduced matrix below.
[ 0 1 | 7 ]
Show solution
The first row means x = −4 and the second means y = 7. The system solution is (−4, 7).
Solve using row operations
Solve x + y = 5 and 2x − y = 4 using an augmented matrix.
Show solution
[ 2 −1 | 4 ]
R₂ → R₂ − 2R₁:
[ 0 −3 | −6 ]
R₂ → (−1/3)R₂, then R₁ → R₁ − R₂:
[ 0 1 | 2 ]
Solution: (3, 2).
Lesson 4 — Feasible Regions & Objectives
Describe a feasible region
Graph conceptually: x ≥ 0, y ≥ 0, x + y ≤ 6. What are the vertices?
Show solution
The region is the triangle in the first quadrant beneath x + y = 6. Its vertices are (0, 0), (6, 0), and (0, 6).
Test an objective function
For the vertices (0, 0), (6, 0), and (0, 6), maximize P = 4x + 7y.
Show solution
The maximum is 42 at (0, 6).
Lesson 5 — Applications & Linear Programming
Translate a constraint
A robot team can use at most 18 total motors. Robot A uses 2 motors and Robot B uses 3. If x and y are the numbers built, write the constraint.
Show solution
Because x and y count robots, also include x ≥ 0 and y ≥ 0, and use whole numbers in the final interpretation.
Build an objective
Each Robot A earns 6 competition points and each Robot B earns 10. Write the objective function to maximize score.
Show solution
Linear-programming decision
A lab can run experiment A or B. A uses 2 minutes of sensor time and 1 minute of setup; B uses 1 minute of sensor time and 2 minutes of setup. At most 12 minutes of each resource are available. A earns 4 data-quality points and B earns 5. What integer combination maximizes the score?
Show solution
x + 2y ≤ 12
x ≥ 0, y ≥ 0
The two main boundary lines intersect where 2x + y = 12 and x + 2y = 12, giving x = 4 and y = 4. The other vertices are (6, 0) and (0, 6). Test S = 4x + 5y:
The best choice is 4 A trials and 4 B trials, for 36 points.
Equations & Rules
Every formula and rule for the unit on one printable page.
System solution
A value or ordered pair that makes every equation true at the same time.
Independent system
Dependent system
Inconsistent system
Substitution
Solve one equation for a variable, substitute its equivalent expression into the other equation, then back-substitute.
Elimination
Create opposite coefficients, add the equations, solve the remaining one-variable equation, then back-substitute.
Augmented matrix
Row operations
Swap rows; multiply a row by a nonzero scalar; add a multiple of one row to another.
Inequality boundary
Solid for ≤ or ≥; dashed for < or >.
Feasible region
The overlap satisfying all constraints simultaneously.
Objective function
Vertex principle
For a linear objective on a closed polygonal feasible region, test the vertices for the maximum or minimum.
Unit 2 Review Guide
The strongest sign that you understand systems is that you can choose a useful method instead of forcing every problem into the same procedure.
Solving systems
- Graph two linear equations and identify their intersection.
- Use substitution efficiently when one variable is isolated or easy to isolate.
- Use elimination by creating opposite coefficients.
- Check an ordered-pair solution in both original equations.
- Recognize independent, dependent, and inconsistent systems.
Matrices & optimization
- Create an augmented matrix with coefficients in a consistent order.
- Use and explain legal row operations.
- Interpret a matrix in reduced row-echelon form.
- Graph multiple inequalities and identify the common feasible region.
- Find vertices as intersections of boundary lines.
- Write and evaluate objective functions.
- Interpret a maximum or minimum in the original context.
Method Choice
| Situation | Usually a good method | Why |
|---|---|---|
| Both equations are already easy to graph | Graphing | The intersection may be visible immediately and gives geometric meaning. |
| One variable is already isolated | Substitution | You can replace it directly in the other equation. |
| Coefficients are opposites or simple multiples | Elimination | One variable can disappear in very few steps. |
| Several equations or variables, or a calculator-based solution | Matrices | Compact organization and systematic row reduction. |
| Several limits must all be satisfied | Inequality system | The overlap shows every allowed choice. |
| Need best profit, cost, output, or efficiency | Linear programming | Test an objective at the vertices of the feasible region. |
Before you turn in the assessment
- Every answer written as an ordered pair, not just the first variable you solved for.
- Each candidate solution checked in both original equations.
- 0 = 0 read as dependent and 0 = 7 read as inconsistent, not as an arithmetic error.
- Fractional intersections accepted rather than rounded away.
- Augmented matrices written with variables in the same order in every row, and zeros for missing terms.
- Boundaries drawn solid for ≤ and ≥, dashed for < and >.
- Optimization answers found by testing vertices, not interior points.
- Every applied answer interpreted in context, with units and with whole numbers when the quantity is countable.
Performance Task: Collision Course
A hands-on physics task that predicts exactly when and where two moving objects meet — then redesigns the setup under real constraints.
Predict the Meeting Point
Driving question: Can a mathematical system predict exactly when and where two moving objects will meet — and can you redesign the setup under real constraints to make the meeting happen where you want?
Students measure the approximately constant velocities of two toy cars or dynamics carts moving along the same line. They build one position function for each object, solve the two-function system in several ways, test the predicted meeting experimentally, and then complete a constrained engineering redesign.
Materials
- Two constant-speed toy cars or dynamics carts
- 3–6 m straight track or floor lane
- Measuring tape or meter sticks
- Masking tape for coordinate marks
- Stopwatch or phone video
- Graph paper or spreadsheet
Physics connection
For constant velocity, each object’s position changes linearly with time:
x₂(t) = v₂t + x₂₀
The objects meet when their positions are equal. That physical event is exactly the intersection of two linear equations.
Part A — Calibrate each moving object
- Choose a coordinate axis along the track and mark x = 0.
- Run Cart A alone. Measure position at at least five times and create a linear model x₁(t).
- Run Cart B alone in the direction it will use in the collision trial. Measure position at at least five times and create x₂(t).
- Interpret each slope as a velocity, including its sign, and each intercept as a starting position.
Part B — Predict the meeting
- Choose starting positions so both carts remain on the safe track area.
- Graph x₁(t) and x₂(t) on the same axes and estimate the intersection.
- Solve the system by substitution.
- Rewrite the equations in standard form and solve the same system by elimination.
- Create the 2×3 augmented matrix and reduce it by hand or calculator.
- All methods should identify approximately the same ordered pair (time, position). Explain why.
Part C — Experimental test
- Place both carts at the modeled starting positions.
- Start them simultaneously and record the run on video when possible.
- Measure the actual meeting time and position.
- Calculate the absolute error for both time and position.
- Identify at least two physical reasons the real result differs from the algebraic prediction.
Part D — Engineering constraints
Now treat the experiment as a design problem. Let a be the number of 0.25 m starting-position adjustments made to Cart A and b the number made to Cart B. Your teacher may modify the limits to fit the room. Use these sample constraints:
a + b ≤ 10
2a + b ≤ 14
a + 3b ≤ 18
Interpret each constraint as a physical restriction — total available adjustment distance, limited setup time, or restricted track space. Graph the feasible region and determine its vertices.
Part E — Optimize a design choice
Suppose each adjustment to Cart A improves predicted target accuracy by 3 design points and each adjustment to Cart B improves it by 4 points:
- Evaluate Q at every feasible vertex.
- Select the maximum feasible design.
- If a vertex contains fractional values, explain whether it is physically meaningful and choose a nearby feasible integer point if needed.
- Apply your chosen adjustment to the physical setup, rerun the experiment, and compare the new meeting point with the original trial.
Required submission — the model
- Calibration data for both carts
- Two position-time equations with units
- Combined position-time graph
- Graphical intersection estimate
- Substitution solution
- Elimination solution
- Augmented matrix and RREF result
Required submission — the test and redesign
- Experimental meeting time and position
- Error calculations and discussion
- Constraint equations with meanings
- Feasible-region graph and vertices
- Objective-function calculations
- Chosen optimized setup
- Final reflection on model limitations
Suggested Rubric — 24 points
| Category | 4 — Strong | 3 — Proficient | 2 — Developing | 1 — Incomplete |
|---|---|---|---|---|
| Motion data & models | Reliable data, correct units, and two defensible linear models. | Models are usable with only minor data or notation issues. | Models show repeated errors or weak data. | Insufficient data or models. |
| System methods | Graphing, substitution, elimination, and matrix methods all agree and are clearly shown. | All methods attempted with one minor error. | Only two or three methods are correct. | Little evidence of a valid system solution. |
| Experimental verification | Meeting point is tested; time and position errors are quantified and explained. | Prediction is tested with basic error analysis. | Test completed but analysis is limited. | No meaningful verification. |
| Constraints & feasible region | Constraints are correctly graphed and interpreted, and all relevant vertices are identified. | Feasible region is mostly correct with minor issues. | Partial region or unclear constraint meanings. | Constraints not correctly represented. |
| Optimization | Objective is correctly evaluated at vertices and a physically valid optimum is justified. | Correct decision with a small calculation or interpretation issue. | Partial vertex testing or weak justification. | No defensible optimum. |
| Physics & conclusion | Clearly connects slope to velocity, intersection to shared position and time, and model error to real motion. | Connections are correct but brief. | Some correct physics language but limited synthesis. | Conclusion unsupported or physically incorrect. |
Textbook Basis
Where this unit comes from, and what was rewritten for the web.
This page is an instructional adaptation of Chapter 2, Systems of Linear Equations, from the W.A. Algebra 2 (Summer 2016) CK-12 FlexBook. Lesson sequence and learning targets follow sections 2.1–2.5 of the source text; worked and practice examples on this page have been rewritten or newly created for web instruction.
CK-12 Foundation content is provided under the Creative Commons Attribution-NonCommercial 3.0 license unless otherwise noted. Visit CK-12 for platform and licensing information.
Where to Go Next
Unit 2 takes the single lines of Unit 1 and asks what happens when several conditions must hold at once.
Unit 1: Linear Toolkit
Equations and inequalities, slope, the three forms of a line, and function notation — the skills every method in this unit depends on.
Back to Unit 1 →All Algebra 2 Units
The full Algebra 2 course map, with every content summary, sample problem set, equation sheet, and performance task in one place.
Back to Algebra 2 →Physics: Kinematics
The performance task predicts where two moving objects meet. The physics side of the same idea, including the graphs and the vocabulary.
Open Kinematics →