Chapter 4: Systems of Equations and Ineq. Lecture notes Math 1010 Section 4.1: Systems of Equations Systems of equations A system of equations consists of two or more equations involving two or more variables ax + by = c dx + ey = f A solution of such a system is an ordered pair (x, y) of real numbers that satisfies each equation in the system. The solutions of the system correspond to the points of intersection of the graphs of the equations. A system of linear equations can have exactly one solution, infinitely many solutions, or no solution. Ex.1 Checking solutions of a system of equations. Check whether each ordered pair is a solution of the system of equations x + y =6 2x - 5y = -2 (1) (3, 3) (2) (4, 2) 1
Transcript
Chapter 4: Systems of Equations and Ineq. Lecture notes Math 1010
Section 4.1: Systems of Equations
Systems of equationsA system of equations consists of two or more equations involving two or more variables{
ax+ by = cdx+ ey = f
A solution of such a system is an ordered pair (x, y) of real numbers that satisfies each equation in thesystem. The solutions of the system correspond to the points of intersection of the graphs of the equations.A system of linear equations can have exactly one solution, infinitely many solutions, or no solution.
Ex.1 Checking solutions of a system of equations.Check whether each ordered pair is a solution of the system of equations{
x+ y = 62x− 5y = −2
(1) (3, 3)(2) (4, 2)
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Chapter 4: Systems of Equations and Ineq. Lecture notes Math 1010
Ex.2Use the graphical method to solve the system of equations{
2x+ 3y = 72x− 5y = −1
The method of substitutionIn order to solve a system of two equations involving two variables you need to follow these steps:
(1) Solve one of the equations for one variable in terms of the other.(2) Substitute the expression obtained in Step 1 in the other equation to obtain an equation in one
variable.(3) Solve the equation obtained in Step 2.(4) Back-substitute the solution from Step 3 in the expression obtained in Step 1 to find the value of the
other variable.(5) Check the solution to see that it satisfies both of the original equations.
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Chapter 4: Systems of Equations and Ineq. Lecture notes Math 1010
Ex.3 Solve the following system of equations{−x+ y = 33x+ y = −1
Ex.4Solve the following system of equations {
2x− 2y = 0x− y = 1
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Chapter 4: Systems of Equations and Ineq. Lecture notes Math 1010
Ex.5Find two positive integers such that the sum of the numbers is 8 and their difference is 2.
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Chapter 4: Systems of Equations and Ineq. Lecture notes Math 1010
Section 4.2: Linear Systems in Two Variables
The method of eliminationThe steps for solving a system of linear equations by the method of elimination are the following:
(1) Obtain coefficients for x (or y) that are opposites by multiplying all terms of one or both equationsby suitable constants.
(2) Add the equations to eliminate one variable, and solve the resulting equation.(3) Back-substitute the value obtained in Step 2 in either the original equations and solve for the other
variable.(4) Check your solution in both of the original equations.
Ex.1Solve the following system of linear equations{
3x+ 2y = 45x− 2y = 8
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Chapter 4: Systems of Equations and Ineq. Lecture notes Math 1010
Ex.2Solve the following system of linear equations{
4x− 5y = 133x− y = 7
Ex.3Solve the following system of linear equations{
3x+ 9y = 82x+ 6y = 7
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Chapter 4: Systems of Equations and Ineq. Lecture notes Math 1010
Ex.4Solve the following system of linear equations{
−2x+ 6y = 34x− 12y = −6
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Chapter 4: Systems of Equations and Ineq. Lecture notes Math 1010
Section 4.3: Linear Systems in Three Variables
Ex.1Solve the following system of linear equations x− 2y + 2z = 9
y + 2z = 5z = 3
The method of Gauss eliminationTwo systems of equations are equivalent if they have the same solution set. To solve a system that is not inrow-echelon form, first convert it to an equivalent system that is in row-echelon form and then solve thenew system.Each of the following row operations on a system of linear equations produces an equivalent system oflinear equations.
(1) Interchange two equations.(2) Multiply one of the equations by a nonzero constant.(3) Add a multiple of one of the equations to another equation to replace the latter equation.
Ex.2Solve the following system of linear equations{
3x− 2y = −1x− y = 0
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Chapter 4: Systems of Equations and Ineq. Lecture notes Math 1010
Ex.3Solve the following system of linear equations x− 2y + 2z = 9
−x+ 3y = −42x− 5y + z = 10
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Chapter 4: Systems of Equations and Ineq. Lecture notes Math 1010
Ex.4Solve the following system of linear equations 4x+ y − 3z = 11
2x− 3y + 2z = 9x+ y + z = −3
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Chapter 4: Systems of Equations and Ineq. Lecture notes Math 1010
Ex.5Solve the following system of linear equations x− 3y + z = 1
2x− y − 2z = 2x+ 2y − 3z = −1
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Chapter 4: Systems of Equations and Ineq. Lecture notes Math 1010
Ex.6Solve the following system of linear equations x+ y − 3z = −1
y − z = 0−x+ 2y = 1
Solutions of a linear systemFor a system of linear equations, exactly one of the following is true.
(1) There is exactly one solution.(2) There are infinitely many solutions.(3) There is no solution.
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Chapter 4: Systems of Equations and Ineq. Lecture notes Math 1010
Section 4.4: Matrices and Linear Systems
MatricesA matrix (plural matrices) is a rectangular array of real numbers, such as 3 1 −1
2 5 02 4 1
An item in a matrix is called an entry or an element. If the matrix has n rows and m columns, its order isn×m. A matrix with the same number of rows and columns is called a square matrix.
Ex.1Determine the order of each matrix.
(1) (1 2 40 5 −3
)(2) (
0 00 0
)(3) 1 −1
5 04 −2
Augmented matricesA matrix derived from a system of linear equations is the augmented matrix. The matrix derived from thecoefficients of the system (but not including the constant terms) is the coefficient matrix of the system.
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Chapter 4: Systems of Equations and Ineq. Lecture notes Math 1010
Ex.2Form the coefficient matrix and the augmented matrix for each system.
(1) {−x+ 5y = 27x− 2y = −6
(2) 3x+ 2y − z = 1x+ 2z = −3−2x− y = 4
Ex.3Write the system of linear equations represented by each matrix.
(1) 3 −5... 4
−1 2... 0
(2) 1 3
... 2
0 1... −3
(3)
2 0 −8... 1
−1 1 1... 2
5 −1 7... 3
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Chapter 4: Systems of Equations and Ineq. Lecture notes Math 1010
Elementary row operationsAs in the case of systems of linear equations there are three elementary row operations that produce equivalentmatrices. The elementary row operations are
(1) Interchange two rows.(2) Multiply a row by a nonzero constant.(3) Add a multiple of a row to another row.
Ex.4Given the matrix (
1 2 40 5 −3
)do the following elementary row operations
(1) Interchange the first and second row.(2) Multiply the first row by 3.(3) Add −2 times the first row to the second row.
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Chapter 4: Systems of Equations and Ineq. Lecture notes Math 1010
Definition of row-echelon form of a matrixA matrix is in row-echelon form if
(1) All entries at the bottom of the diagonal are equal to zero.(2) The first nonzero entry in a row is 1 (leading 1).(3) For two consecutive (nonzero) rows, the leading 1 in the higher row is farther to the left than the
leading 1 in the lower row.
Ex.5The matrix 1 2 4
0 1 −30 0 1
is in row-echelon form.
Gaussian eliminationWe can perform the Gaussian elimination using matrices and the following steps:
(1) Write the augmented matrix of the system of linear equations.(2) Use elementary row operations to rewrite the augmented matrix in row-echelon form.(3) Write the system of linear equations corresponding to the matrix in row-echelon form, and use
back-substitution to find the solution.
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Chapter 4: Systems of Equations and Ineq. Lecture notes Math 1010
Ex.6Solve the following system of linear equations{
2x− 3y = −2x+ 2y = 13
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Chapter 4: Systems of Equations and Ineq. Lecture notes Math 1010
Ex.7Solve the following system of linear equations 3x+ 3y = 9
2x− 3z = 106y + 4z = −12
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Chapter 4: Systems of Equations and Ineq. Lecture notes Math 1010
Ex.8Solve the following system of linear equations{
6x− 10y = −49x− 15y = 5
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Chapter 4: Systems of Equations and Ineq. Lecture notes Math 1010
Ex.9Solve the following system of linear equations{
12x− 6y = −3−8x+ 4y = 2
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Chapter 4: Systems of Equations and Ineq. Lecture notes Math 1010
Section 4.5: Determinants and Linear Systems
Determinant of a 2× 2 matrixGiven a matrix 2× 2
A =
(a1 b1a2 b2
)the determinant is
det(A) =
∥∥∥∥ a1 b1a2 b2
∥∥∥∥ = a1b2 − a2b1
Ex.1Find the determinant of each matrix.
(1) (2 −31 4
)(2) (
−1 22 −4
)(3) (
1 32 5
)
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Chapter 4: Systems of Equations and Ineq. Lecture notes Math 1010
Determinant of a 3× 3 matrixGiven a matrix 3× 3
A =
a1 b1 c1a2 b2 c2a3 b3 c3
the determinant is
det(A) = a1
∥∥∥∥ b2 c2b3 c3
∥∥∥∥− b1
∥∥∥∥ a2 c2a3 c3
∥∥∥∥+ c1
∥∥∥∥ a2 b2a3 b3
∥∥∥∥
Ex.2Find the determinant of each matrix.
(1) −1 1 20 2 33 4 2
(2) 1 2 1
3 0 24 0 −1
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Chapter 4: Systems of Equations and Ineq. Lecture notes Math 1010
Cramer’s ruleFor the system of linear equations {
a1x+ b1y = c1a2x+ b2y = c2
the solution is given by
x =Dx
D=
∥∥∥∥ c1 b1c2 b2
∥∥∥∥∥∥∥∥ a1 b1a2 b2
∥∥∥∥y =
Dy
D=
∥∥∥∥ a1 c1a2 c2
∥∥∥∥∥∥∥∥ a1 b1a2 b2
∥∥∥∥where D 6= 0.For the system of linear equations a1x+ b1y + c1z = d1
a2x+ b2y + c2z = d2a3x+ b3y + c3z = d3
the solution is given by
x =Dx
D=
∥∥∥∥∥∥d1 b1 c1d2 b2 c2d3 b3 c3
∥∥∥∥∥∥∥∥∥∥∥∥a1 b1 c1a2 b2 c2a3 b3 c3
∥∥∥∥∥∥
y =Dy
D=
∥∥∥∥∥∥a1 d1 c1a2 d2 c2a3 d3 c3
∥∥∥∥∥∥∥∥∥∥∥∥a1 b1 c1a2 b2 c2a3 b3 c3
∥∥∥∥∥∥
z =Dz
D=
∥∥∥∥∥∥a1 b1 d1a2 b2 d2a3 b3 d3
∥∥∥∥∥∥∥∥∥∥∥∥a1 b1 c1a2 b2 c2a3 b3 c3
∥∥∥∥∥∥where D 6= 0.
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Chapter 4: Systems of Equations and Ineq. Lecture notes Math 1010
Ex.3Use Cramer’s rule to solve the following system of linear equations{
4x− 2y = 103x− 5y = 11
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Chapter 4: Systems of Equations and Ineq. Lecture notes Math 1010
Ex.4Use Cramer’s rule to solve the following system of linear equations −x+ 2y − 3z = 1
2x+ z = 03x− 4y + 4z = 2
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Chapter 4: Systems of Equations and Ineq. Lecture notes Math 1010
Section 4.6: Systems of Linear Inequalities
Graphing a system of linear inequalities
(1) Sketch the line that corresponds to each inequality.(2) Lightly shade the half-plane that is the graph of each linear inequality.(3) The graph of the system is the intersection of the half-planes.
Ex.1Sketch the graph of the following system of linear inequalities{
2x− y ≤ 5x+ 2y ≥ 2
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Chapter 4: Systems of Equations and Ineq. Lecture notes Math 1010
Ex.2Sketch the graph of the following system of linear inequalities{
y < 4y > 1
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Chapter 4: Systems of Equations and Ineq. Lecture notes Math 1010
Ex.3Sketch the graph of the following system of linear inequalities x− y < 2
x > −2y ≤ 3
Ex.4Sketch the graph of the following system of linear inequalities
x+ y ≤ 53x+ 2y ≤ 12
x ≥ 0y ≥ 0
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Chapter 4: Systems of Equations and Ineq. Lecture notes Math 1010
Ex.5Find a system of inequalities that defines the region shown below.
