Systems of Equations and Inequalities
Systems of Linear Equations in Two Variables
System of Equations
A system of equations is a set of equations that involve the same variables. A system of linear equations is one in which every equation is linear. A solution of a system is an assignment of values to the variables that makes every equation true. To solve a system means to find all of its solutions. Graphically, each linear equation is a line, and a solution is a point lying on every line in the system — a point of intersection.
Substitution Method
A strategy for solving a system in which you isolate one variable in one equation and substitute the resulting expression into the other equation, reducing the system to a single equation in one variable. Step 1 — Solve for one variable in terms of the other. Step 2 — Substitute that expression into the other equation and solve. Step 3 — Back-substitute the value found into the expression from Step 1 to get the remaining variable.
Elimination Method
A strategy that combines the equations (by adding or subtracting) so one variable cancels. Step 1 — Adjust the coefficients by multiplying equations so that the coefficient of one variable in one equation is the negative of its coefficient in the other. Step 2 — Add the equations to eliminate that variable and solve. Step 3 — Back-substitute into an original equation to find the remaining variable.
Graphical Method
Using a graphing device, express each equation by solving for \(y\), graph both on the same screen, and read off the coordinates of the point(s) of intersection. Useful for approximate solutions with awkward coefficients.
Number of Solutions of a Linear System in Two Variables
Because two lines either cross once, run parallel, or coincide, exactly one of the following is always true: the system has exactly one solution, no solution, or infinitely many solutions. A system with no solution is inconsistent; a system with infinitely many solutions is dependent.
Modeling with Linear Systems
Guidelines for translating a word problem into a system: (1) identify the variables (what the problem asks you to find), (2) express all unknown quantities in terms of those variables, (3) set up a system that captures the relationships in the problem, and (4) solve the system, check, and state the answer in a sentence.
(Form used to express the infinitely many solutions of a dependent two-variable system, as in Example 6: the ordered pair \(\left(t, \tfrac{1}{2}t - 2\right)\) for any real \(t\).)
Systems of Linear Equations in Several Variables
Linear Equation in n Variables
An equation that can be written as \( a_1 x_1 + a_2 x_2 + \cdots + a_n x_n = c \), where the \(a_i\) and \(c\) are real numbers. Equations are linear when each term contains at most one variable to the first power — no squares, square roots, or products of variables. With three or four variables we usually write \(x, y, z, w\) instead of \(x_1, x_2, x_3, x_4\).
Triangular Form & Back-Substitution
A system is in triangular form when each successive equation drops a variable (the second equation has no \(x\), the third has neither \(x\) nor \(y\), etc.). Such systems are solved easily by back-substitution: solve the last equation, then work upward, substituting known values as you go.
Gaussian Elimination
The process of using equation operations to convert a linear system into an equivalent triangular system, which is then solved by back-substitution. Two systems are equivalent when they have the same solutions.
Number of Solutions / Parameters
A linear system has exactly one solution, no solution, or infinitely many solutions. No solution means it is inconsistent (Gaussian elimination yields a false statement). Infinitely many means it is dependent; the solution is described using one or more parameters — free variables that may take any real value.
1. Add a nonzero multiple of one equation to another.
2. Multiply an equation by a nonzero constant.
3. Interchange the positions of two equations.
Matrices and Systems of Linear Equations
Matrix and Dimension
A matrix is a rectangular array of numbers. An \(m \times n\) matrix has \(m\) rows and \(n\) columns; we say it has dimension \(m \times n\). The number \(a_{ij}\) (the entry in row \(i\), column \(j\)) carries a double subscript indicating its position.
Augmented Matrix
The augmented matrix of a linear system records only the coefficients and the constants of the system, with a vertical bar separating the coefficient columns from the constant column. A missing variable corresponds to a \(0\) entry.
Elementary Row Operations
Operations on the rows of an augmented matrix that mirror the equation operations and do not change the solution set: (1) add a multiple of one row to another, (2) multiply a row by a nonzero constant, (3) interchange two rows. Notation: \(R_i + kR_j \to R_i\), \(kR_i\), and \(R_i \leftrightarrow R_j\).
Row-Echelon & Reduced Row-Echelon Form
A matrix is in row-echelon form when: (1) the first nonzero number in each row (the leading entry) is \(1\); (2) each leading entry sits to the right of the one above it; (3) all-zero rows are at the bottom. It is in reduced row-echelon form if, additionally, (4) every entry above and below each leading \(1\) is \(0\).
Gaussian vs. Gauss-Jordan Elimination
Gaussian elimination reduces the augmented matrix to row-echelon form and finishes with back-substitution. Gauss-Jordan elimination continues to reduced row-echelon form, so the solution can be read directly with no back-substitution. (TI-83/84 commands: ref and rref.)
Leading Variables; Inconsistent & Dependent Systems
A leading variable corresponds to a leading entry in the row-echelon form. From the row-echelon form: a row of the form \(0 = c\) (\(c \neq 0\)) means no solution (inconsistent); if every variable is leading, there is exactly one solution; if some variables are non-leading and the system is not inconsistent, there are infinitely many solutions (dependent), expressed with non-leading variables as parameters. If a system in row-echelon form has \(n\) nonzero equations in \(m\) variables (\(m > n\)), the solution has \(m - n\) non-leading variables.
rref): \( x = 5,\; y = 2,\; z = 10 \).rref) is cleaner since the answer falls out directly. Watch the bottom row: \(0=c\,(c\neq0)\) ⇒ no solution; \(0=0\) ⇒ infinitely many. This same row-reduction is the workhorse behind solving the large stiffness/circuit systems you'll meet in mechanics and electrical courses.The Algebra of Matrices
Equality of Matrices
Two matrices are equal when they have the same dimension and every pair of corresponding entries is equal.
Addition, Subtraction, and Scalar Multiplication
Matrices of the same dimension are added or subtracted entry by entry (otherwise the operation is undefined). To multiply a matrix by a number (a scalar), multiply every entry by that number — the scalar product.
Matrix Multiplication
The product \(AB\) is defined only when the number of columns of \(A\) equals the number of rows of \(B\): if \(A\) is \(m \times n\) and \(B\) is \(n \times k\), then \(AB\) is \(m \times k\). The entry \(c_{ij}\) of \(AB\) is the inner product of the \(i\)th row of \(A\) with the \(j\)th column of \(B\). Matrix multiplication is not commutative: in general \(AB \neq BA\).
Stochastic Matrix
A matrix whose columns each sum to \(1\). Stochastic matrices arise frequently in statistics and demographic modeling, where each column distributes proportions that must total \(100\%\).
Computer Graphics (Application)
Digital images are stored as matrices: each pixel gets a value (e.g. \(0\)–\(255\) on a gray scale). Operations on the matrix transform the image — adding a constant darkens it, subtracting lightens it, and raising the dark values while lowering the light ones increases contrast.
Inverses of Matrices and Matrix Equations
Identity Matrix
The identity matrix \(I_n\) is the \(n \times n\) matrix with \(1\)s on the main diagonal and \(0\)s elsewhere. It behaves like the number \(1\): \( A\,I_n = A \) and \( I_n\,B = B \) whenever the products are defined. A square matrix has equal numbers of rows and columns; its main diagonal runs from top-left to bottom-right.
Inverse of a Matrix
For a square \(n \times n\) matrix \(A\), the inverse \(A^{-1}\) (when it exists) satisfies \( AA^{-1} = A^{-1}A = I_n \). It plays the role of a reciprocal. A matrix with no inverse is called singular. For a \(2 \times 2\) matrix, the quantity \(ad - bc\) is the determinant; if it is \(0\), no inverse exists.
Finding the Inverse of an n × n Matrix
Form the \(n \times 2n\) matrix \([\,A \mid I_n\,]\) and use elementary row operations to reduce the left half to the identity. The right half is then transformed automatically into \(A^{-1}\). If a row of zeros appears on the left, \(A\) has no inverse.
Matrix Equations
A linear system \(AX = B\) (with \(A\) the coefficient matrix, \(X\) the variable matrix, \(B\) the constants) is solved by multiplying on the left by \(A^{-1}\): \( X = A^{-1}B \), provided \(A^{-1}\) exists. This is especially efficient when several systems share the same coefficient matrix — \(A^{-1}\) is found once and reused.
Determinants and Cramer's Rule
Determinant
A single number assigned to every square matrix, written \(\det(A)\) or \(|A|\). For a \(1\times1\) matrix \([a]\), \(\det = a\). The determinant is central to deciding invertibility and to Cramer's Rule.
Minors and Cofactors
The minor \(M_{ij}\) of entry \(a_{ij}\) is the determinant of the matrix left after deleting row \(i\) and column \(j\). The cofactor is \( A_{ij} = (-1)^{i+j}M_{ij} \) — the minor with a sign from the checkerboard pattern \(\begin{smallmatrix}+&-&+\\-&+&-\\+&-&+\end{smallmatrix}\).
Expanding a Determinant
The determinant of an \(n\times n\) matrix is obtained by multiplying each entry of the first row by its cofactor and summing. In fact you may expand along any row or column and get the same value — choose one with many zeros to save work.
Invertibility Criterion
A square matrix \(A\) has an inverse if and only if \(\det(A) \neq 0\). This is the reason for the name "determinant" — it determines whether the inverse exists.
Row and Column Transformations
Adding a multiple of one row to another row (or one column to another column) does not change the determinant. Use this to introduce zeros before expanding, greatly reducing the arithmetic.
Cramer's Rule
A method that expresses the solution of a square linear system \(DX = B\) (with \(\det(D) \neq 0\)) as ratios of determinants. Convenient for small systems but impractical for large ones, and inapplicable when \(\det(D) = 0\).
where \(D_{x_i}\) is \(D\) with its \(i\)th column replaced by the constants \(B\).
choosing the sign that makes the area positive.
Partial Fractions
Partial Fraction Decomposition
The reverse of combining fractions over a common denominator: a rational function \( r(x) = \dfrac{P(x)}{Q(x)} \) (with \(\deg P < \deg Q\)) is written as a sum of simpler fractions called partial fractions. Every polynomial with real coefficients factors completely into linear factors \(ax + b\) and irreducible quadratic factors \(ax^2 + bx + c\), and each factor contributes a term to the decomposition.
Case 1 — Distinct Linear Factors
When \(Q(x)\) is a product of distinct linear factors, each factor \(a_i x + b_i\) contributes one term \(\dfrac{A_i}{a_i x + b_i}\).
Case 2 — Repeated Linear Factors
When a linear factor \(ax + b\) is repeated \(k\) times, it contributes \(k\) terms, one for each power up to \(k\): \(\dfrac{A_1}{ax+b} + \dfrac{A_2}{(ax+b)^2} + \cdots + \dfrac{A_k}{(ax+b)^k}\).
Case 3 — Distinct Irreducible Quadratic Factors
Each irreducible quadratic factor \(ax^2 + bx + c\) contributes a term with a linear numerator: \(\dfrac{Ax + B}{ax^2 + bx + c}\).
Case 4 — Repeated Irreducible Quadratic Factors
An irreducible quadratic raised to the \(k\)th power contributes \(k\) terms, one for each power, each with a linear numerator.
Improper Rational Functions
These techniques require \(\deg P < \deg Q\). If the numerator's degree is not smaller, first use long division to write \(r(x)\) as a polynomial plus a proper remainder fraction, then decompose the remainder.
Systems of Nonlinear Equations
Nonlinear Systems
Systems in which the equations are not all linear. The same tools from §10.1 — substitution and elimination — still apply, and the graphical method is often especially useful because the graphs (circles, parabolas, etc.) can intersect in several points.
Graphical Interpretation
The solutions of a nonlinear system are the intersection points of the graphs. Because curves can cross more than once, a nonlinear system may have several solutions — and curves that appear close may not actually intersect, so checking algebraically (or zooming carefully) matters.
Solve by the same methods as linear systems:
• Substitution — solve one equation for a variable, substitute into the other.
• Elimination — scale and add equations to cancel a term.
• Graphical — graph each curve and locate intersection points.
Systems of Inequalities
Graph of an Inequality
The graph of an inequality is a region of the plane whose boundary is the graph of the corresponding equation. Use a dashed curve for strict inequalities (\(<\) or \(>\)) and a solid curve for \(\leq\) or \(\geq\). A test point not on the boundary tells you which side to shade: if it satisfies the inequality, shade that side.
System of Inequalities
The solution set of a system of inequalities is the set of all points satisfying every inequality — the intersection of the individual solution regions. Graph each inequality on the same axes, shade the overlap, and label the vertices (corner points), found by solving the relevant boundary equations simultaneously. A vertex is part of the solution only if it satisfies all inequalities (open circle if not).
Linear Inequalities
An inequality is linear if it can be written as \(ax + by \leq c\) (or with \(<, >, \geq\)). Systems of linear inequalities bound regions whose edges are line segments.
Bounded vs. Unbounded Regions
A region is bounded if it can be enclosed in a (large enough) circle; otherwise it is unbounded.
Feasible Region (Application)
In applied problems the variables face constraints (limited labor, materials, pollution limits). Expressed as a system of inequalities, the solution set is called the feasible region: its points represent the possible (feasible) combinations of the quantities being studied.
1. Graph the boundary equation (dashed for \(<,>\); solid for \(\leq,\geq\)).
2. Test a point off the boundary; shade the side that satisfies the inequality.
Linear Programming
A modeling technique for finding the optimal allocation of resources — maximizing profit or minimizing cost subject to constraints. It is the most important practical application of systems of linear inequalities. (Kantorovich and Koopmans won the 1975 Nobel Prize in economics for developing it.)
Objective Function, Constraints, Feasible Region
The objective function is the linear expression to be maximized or minimized. The constraints are inequalities limiting the variables; their solution set is the feasible region. The key fact: the objective function attains its maximum and minimum at a vertex of the feasible region — so only the vertices need to be checked.
Guidelines for Linear Programming
(1) Choose the variables — name the quantities \(x\) and \(y\). (2) Find the objective function — write the expression to optimize. (3) Graph the feasible region — express constraints as inequalities and graph their solution. (4) Find the maximum or minimum — evaluate the objective function at each vertex.
A linear objective function attains its maximum and minimum values at the vertices of the feasible region — so evaluate it only at the corner points.