Chapter 7 · Test 2
Systems of Linear Equations for Economics
Substitution, elimination, rank, and economic equilibrium systems.
What you will learn
- Solve a two-equation system
- Solve a linear market system
- Recognize a unique linear solution
Worked example
Solve a two-equation system: follow the method step by step
Try the problem first, then compare your approach with the complete solution and the reasoning behind it.
Solve a two-equation system
Solve 2x + y = 11 and x − y = 1. What is x + y?
2x + y = 11 x − y = 1
In plain English: The same values of x and y must satisfy both linear equations at once.
Bold blue terms in this symbol guide include definitions. Hover, focus, or tap a term. Each row also shows how to read the notation aloud.
- x, y
- the two unknown variablesHow to read: x and y
Solution
- Add the two equations: (2x + y) + (x − y) = 11 + 1.
- Simplify: 3x = 12, so x = 4.
- Substitute into x − y = 1: 4 − y = 1.
- Solve: y = 3.
- Therefore x + y = 4 + 3 = 7.
Logic and solving tips
- Elimination combines equations so one unknown disappears, leaving a one-variable equation.
- Choose an addition or subtraction that cancels one unknown, then substitute back to recover the other.
Where economists use it
An economist solves a system of market-clearing and accounting equations to recover several unknown prices, quantities, or sector balances at once.
Key idea: Elimination combines equations so one unknown disappears, leaving a one-variable equation.
Applied case study
Solving a two-sector production plan
Scenario
Input-output economists solve simultaneous equations because gross output must cover intermediate use and final demand. BEA defines a direct-requirements coefficient as the input purchased directly per dollar of output. This example compresses the economy into teaching agriculture and manufacturing sectors.
Problem
With A = [[0.2, 0.2], [0.1, 0.4]] and d = [60, 50]ᵀ, solve x = Ax + d by elimination. Outputs are in billions of dollars.
0.8x_A - 0.2x_M = 60 -0.1x_A + 0.6x_M = 50
In plain English: The same unknown values must satisfy every equation in the system at once, so the equations are solved jointly.
Worked solution
- Scale the equations to obtain 4x_A - x_M = 300 and -x_A + 6x_M = 500.
- Multiply the first equation by 6: 24x_A - 6x_M = 1800.
- Add the second equation: 23x_A = 2300, so x_A = 100.
- Substitute into 4x_A - x_M = 300 to get x_M = 100.
Economic interpretation: The teaching economy requires $100 billion of gross output in each sector to meet intermediate use and final demand.
Source: U.S. Bureau of Economic Analysis, Requirements Tables FAQ BEA supplies the input-output application and the meaning of requirements coefficients. The two-sector aggregation, coefficients, and final-demand values are illustrative, not extracted from a BEA table.
Reasoning habits that help
- Elimination combines equations so one unknown disappears, leaving a one-variable equation.
- A market-clearing model is a system because price and quantity must satisfy demand, supply, and equilibrium simultaneously.
- Nonsingularity, a nonzero determinant, invertibility, and uniqueness are equivalent for a square coefficient matrix.
Where economists use these methods
- An economist solves a system of market-clearing and accounting equations to recover several unknown prices, quantities, or sector balances at once.
- A market analyst jointly solves demand, supply, and the clearing condition to forecast the equilibrium price and quantity after a policy or cost change.
- Before reporting a forecast, an economist checks whether the model pins down one equilibrium or leaves multiple possible outcomes because an equation is redundant or missing.
Chapter vocabulary
Study 27 key terms
Say what the term means before opening it. Then compare your explanation with the definition, notation, and example.
Linear equation
An equation in which each unknown appears only to the first power and unknowns are not multiplied together.
Textbook reference: Chapter 7, Section 7.1 Solving Systems of Linear Equations, printed pages 235-236System of linear equations
Two or more linear equations whose unknowns must satisfy every equation simultaneously.
Textbook reference: Chapter 7, Section 7.1 Solving Systems of Linear Equations, printed page 236Solution of a linear system
A value for every unknown that makes all equations in the system true at the same time.
Textbook reference: Chapter 7, Section 7.1 Solving Systems of Linear Equations, printed pages 236-237Substitution method
A solution method that isolates an unknown in one equation and substitutes that expression into another.
Textbook reference: Chapter 7, Section 7.1 Solving Systems of Linear Equations, printed pages 239-240Elimination method
A solution method that combines equations so one unknown cancels, reducing the number of unknowns.
Textbook reference: Chapter 7, Section 7.1 Solving Systems of Linear Equations, printed pages 239-240Structural equations
The model's original behavioral, technological, accounting, and equilibrium relationships among endogenous and exogenous variables.
Textbook reference: Chapter 7, Section 7.1 Solving Systems of Linear Equations, printed page 242Reduced forms
Solved equations expressing each endogenous variable only in terms of exogenous variables and parameters.
Textbook reference: Chapter 7, Section 7.1 Solving Systems of Linear Equations, printed pages 242, 245IS curve
The combinations of income and interest rate consistent with equilibrium in the goods market in the chapter's macroeconomic model.
Textbook reference: Chapter 7, Section 7.1 Solving Systems of Linear Equations, printed pages 244-245LM curve
The combinations of income and interest rate consistent with equilibrium in the money market in the chapter's macroeconomic model.
Textbook reference: Chapter 7, Section 7.1 Solving Systems of Linear Equations, printed page 245Overdetermined system
A system with more independent equations than unknowns; it has a solution only if the extra restrictions are mutually consistent.
Textbook reference: Chapter 7, Section 7.1 Solving Systems of Linear Equations, printed pages 247, 249Underdetermined system
A consistent system with fewer independent equations than unknowns, leaving one or more free variables and infinitely many solutions.
Textbook reference: Chapter 7, Section 7.1 Solving Systems of Linear Equations, printed pages 249-250Basic variable
A variable solved in terms of constants and any free variables after row reduction.
Textbook reference: Chapter 7, Section 7.1 Solving Systems of Linear Equations, printed page 250Free variable
A variable not fixed by a pivot equation that may be assigned values to parameterize a system's solutions.
Textbook reference: Chapter 7, Section 7.1 Solving Systems of Linear Equations, printed page 250Row operation
An equation-preserving operation that swaps rows, multiplies a row by a nonzero constant, or adds a multiple of one row to another.
Textbook reference: Chapter 7, Section 7.2 Linear Systems in n-Variables, printed pages 250-251Row reduction
The systematic use of row operations to transform a linear system into an equivalent form that is easier to solve.
Textbook reference: Chapter 7, Section 7.2 Linear Systems in n-Variables, printed pages 250-251Linear dependence
A relationship in which at least one equation can be generated as a linear combination of the others.
Textbook reference: Chapter 7, Section 7.2 Linear Systems in n-Variables, printed pages 252-253Linear independence
The property that no equation in the set is a linear combination of the others.
Textbook reference: Chapter 7, Section 7.2 Linear Systems in n-Variables, printed page 253Consistent system
A system of equations with at least one solution.
Textbook reference: Chapter 7, Section 7.2 Linear Systems in n-Variables, printed page 253Inconsistent system
A system of equations with no solution because its restrictions contradict one another.
Textbook reference: Chapter 7, Section 7.2 Linear Systems in n-Variables, printed pages 253-254Array form
A rectangular display of a system's coefficients and right-hand-side constants, arranged by equations and variables.
Textbook reference: Chapter 7, Section 7.2 Linear Systems in n-Variables, printed pages 254-255Augmented matrix[A | b]
The coefficient matrix of a linear system with its right-hand-side constants appended as an additional column.
How to read: A augmented by b
Textbook reference: Chapter 7, Section 7.2 Linear Systems in n-Variables, printed pages 254-255Reduced row-echelon form
A unique row-reduced matrix form in which each pivot is one, is the only nonzero entry in its column, and pivots move rightward down the rows.
Textbook reference: Chapter 7, Section 7.2 Linear Systems in n-Variables, printed pages 255-256Pivot
The leading one in a nonzero row of a matrix in reduced row-echelon form.
Textbook reference: Chapter 7, Section 7.2 Linear Systems in n-Variables, printed page 256Coefficient matrixA in Ax = b
The matrix containing the coefficients multiplying the unknowns in a linear system.
Textbook reference: Chapter 7, Section 7.2 Linear Systems in n-Variables, printed pages 258-259Homogeneous systemAx = 0
A linear system whose right-hand-side constants are all zero.
Textbook reference: Chapter 7, Section 7.2 Linear Systems in n-Variables, printed pages 259-260Trivial solutionx = 0
The all-zero solution, which every homogeneous linear system possesses.
Textbook reference: Chapter 7, Section 7.2 Linear Systems in n-Variables, printed page 260Nontrivial solution
A solution to a homogeneous system in which at least one unknown is nonzero.
Textbook reference: Chapter 7, Section 7.2 Linear Systems in n-Variables, printed pages 260-261End-of-chapter problems
Practice all 5 Chapter Review Exercises
The index covers every numbered Review Exercise in this chapter of Mathematics for Economics, 3rd edition. Each reference opens the textbook exercise itself, including every listed subpart, with MFE hints, a symbol guide, a worked explanation, progress tracking, and similar-problem practice.
Practice end-of-chapter problems