Linear Equations Linear systems with two and three unknowns Augmented Matrices Elementary Row operations Section 1.1: Introduction to systems of linear equations Dr K Department of Mathematics and Applied Mathemathematics University of Johannesburg Dr K Introduction to systems of linear equations Linear Equations Linear systems with two and three unknowns Augmented Matrices Elementary Row operations 1 Linear Equations 2 Linear systems with two and three unknowns 3 Augmented Matrices 4 Elementary Row operations Dr K Introduction to systems of linear equations Linear Equations Linear systems with two and three unknowns Augmented Matrices Elementary Row operations Goals: To introduce matrices Develop a way to solve matrices To interpret the solution (if it exists) Dr K Introduction to systems of linear equations Linear Equations Linear systems with two and three unknowns Augmented Matrices Elementary Row operations Linear Equations In two dimensions a line is given by: ax + by = c. In three dimensions a plane is given by: ax + by + cz = d. These are examples of linear equations Generally, we define a linear equation on the n variables x1 , x2 , . . . , xn as a1 x1 + a2 x2 + . . . + an xn = b. where a1 , a2 , . . . , an and b are constants such that not all the a’s are zeros. Dr K Introduction to systems of linear equations Linear Equations Linear systems with two and three unknowns Augmented Matrices Elementary Row operations Linear Equations In two dimensions a line is given by: ax + by = c. In three dimensions a plane is given by: ax + by + cz = d. These are examples of linear equations Generally, we define a linear equation on the n variables x1 , x2 , . . . , xn as a1 x1 + a2 x2 + . . . + an xn = b. where a1 , a2 , . . . , an and b are constants such that not all the a’s are zeros. Dr K Introduction to systems of linear equations Linear Equations Linear systems with two and three unknowns Augmented Matrices Elementary Row operations Linear Equations In two dimensions a line is given by: ax + by = c. In three dimensions a plane is given by: ax + by + cz = d. These are examples of linear equations Generally, we define a linear equation on the n variables x1 , x2 , . . . , xn as a1 x1 + a2 x2 + . . . + an xn = b. where a1 , a2 , . . . , an and b are constants such that not all the a’s are zeros. Dr K Introduction to systems of linear equations Linear Equations Linear systems with two and three unknowns Augmented Matrices Elementary Row operations Linear Equations In the special cases where n = 2 or n = 3, we will often use variables without subscripts and write linear equations as a1 x + a2 y = b a1 x + a2 y + a3 z = b (a1 , a2 not both 0) (a1 , a2 , a3 not all 0) A homogeneous linear equation in the n variables x1 , x2 , . . . , xn can be written as a1 x1 + a2 x2 + . . . + an xn = 0. Dr K Introduction to systems of linear equations Linear Equations Linear systems with two and three unknowns Augmented Matrices Elementary Row operations Linear Equations In the special cases where n = 2 or n = 3, we will often use variables without subscripts and write linear equations as a1 x + a2 y = b a1 x + a2 y + a3 z = b (a1 , a2 not both 0) (a1 , a2 , a3 not all 0) A homogeneous linear equation in the n variables x1 , x2 , . . . , xn can be written as a1 x1 + a2 x2 + . . . + an xn = 0. Dr K Introduction to systems of linear equations Linear Equations Linear systems with two and three unknowns Augmented Matrices Elementary Row operations Linear Equations In the special cases where n = 2 or n = 3, we will often use variables without subscripts and write linear equations as a1 x + a2 y = b a1 x + a2 y + a3 z = b (a1 , a2 not both 0) (a1 , a2 , a3 not all 0) A homogeneous linear equation in the n variables x1 , x2 , . . . , xn can be written as a1 x1 + a2 x2 + . . . + an xn = 0. Dr K Introduction to systems of linear equations Linear Equations Linear systems with two and three unknowns Augmented Matrices Elementary Row operations Identifying linear equations Example Which of the following are linear equations in the variables x, y , z, w, x1 , x2 , x3 ? x − 3y + 4z − 17w = 6 Linear sin x + 7y = 6 Not linear xy + 6z = 4 Not linear −3x 2 Not linear +y =0 5x3 + 4x1 − x2 = 1 Linear πx1 − x2 = 6 Linear Dr K Introduction to systems of linear equations Linear Equations Linear systems with two and three unknowns Augmented Matrices Elementary Row operations Identifying linear equations Example Which of the following are linear equations in the variables x, y , z, w, x1 , x2 , x3 ? x − 3y + 4z − 17w = 6 Linear sin x + 7y = 6 Not linear xy + 6z = 4 Not linear −3x 2 Not linear +y =0 5x3 + 4x1 − x2 = 1 Linear πx1 − x2 = 6 Linear Dr K Introduction to systems of linear equations Linear Equations Linear systems with two and three unknowns Augmented Matrices Elementary Row operations Identifying linear equations Example Which of the following are linear equations in the variables x, y , z, w, x1 , x2 , x3 ? x − 3y + 4z − 17w = 6 Linear sin x + 7y = 6 Not linear xy + 6z = 4 Not linear −3x 2 Not linear +y =0 5x3 + 4x1 − x2 = 1 Linear πx1 − x2 = 6 Linear Dr K Introduction to systems of linear equations Linear Equations Linear systems with two and three unknowns Augmented Matrices Elementary Row operations Identifying linear equations Example Which of the following are linear equations in the variables x, y , z, w, x1 , x2 , x3 ? x − 3y + 4z − 17w = 6 Linear sin x + 7y = 6 Not linear xy + 6z = 4 Not linear −3x 2 Not linear +y =0 5x3 + 4x1 − x2 = 1 Linear πx1 − x2 = 6 Linear Dr K Introduction to systems of linear equations Linear Equations Linear systems with two and three unknowns Augmented Matrices Elementary Row operations Identifying linear equations Example Which of the following are linear equations in the variables x, y , z, w, x1 , x2 , x3 ? x − 3y + 4z − 17w = 6 Linear sin x + 7y = 6 Not linear xy + 6z = 4 Not linear −3x 2 Not linear +y =0 5x3 + 4x1 − x2 = 1 Linear πx1 − x2 = 6 Linear Dr K Introduction to systems of linear equations Linear Equations Linear systems with two and three unknowns Augmented Matrices Elementary Row operations Identifying linear equations Example Which of the following are linear equations in the variables x, y , z, w, x1 , x2 , x3 ? x − 3y + 4z − 17w = 6 Linear sin x + 7y = 6 Not linear xy + 6z = 4 Not linear −3x 2 Not linear +y =0 5x3 + 4x1 − x2 = 1 Linear πx1 − x2 = 6 Linear Dr K Introduction to systems of linear equations Linear Equations Linear systems with two and three unknowns Augmented Matrices Elementary Row operations Identifying linear equations Example Which of the following are linear equations in the variables x, y , z, w, x1 , x2 , x3 ? x − 3y + 4z − 17w = 6 Linear sin x + 7y = 6 Not linear xy + 6z = 4 Not linear −3x 2 Not linear +y =0 5x3 + 4x1 − x2 = 1 Linear πx1 − x2 = 6 Linear Dr K Introduction to systems of linear equations Linear Equations Linear systems with two and three unknowns Augmented Matrices Elementary Row operations Systems of linear equations Definition A system of linear equations or linear system is a finite collection of linear equations. Example x +y −z +w =1 2x − y + 4z = 11 −3x + 4w = −6 The variables are called unknowns. Dr K Introduction to systems of linear equations Linear Equations Linear systems with two and three unknowns Augmented Matrices Elementary Row operations m equations in n unknowns A linear system of m equations in the n unknowns x1 , x2 , . . . , xn can be written as a11 x1 a21 x1 .. . + + a12 x2 a22 x2 .. . + ··· + ··· ··· am1 x1 + am2 x2 + · · · Dr K + + a1n xn a2n xn .. . = = b1 b2 .. . + amn xn = bm Introduction to systems of linear equations Linear Equations Linear systems with two and three unknowns Augmented Matrices Elementary Row operations Solutions to systems of linear equations Definition A solution of a linear system in n unknowns x1 , x2 , . . . , xn is a sequence of n numbers s1 , s2 , . . . , sn for which the substitution x1 = s1 , x2 = s2 , ..., xn = sn makes each equation a true statement. The solution can be written as (s1 , s2 , · · · , sn ) which is called an ordered n-tuple. If n = 2: ordered pair; if n = 3: ordered triple. A linear system is called consistent if it has at least one solution and inconsistent if it has no solution. Dr K Introduction to systems of linear equations Linear Equations Linear systems with two and three unknowns Augmented Matrices Elementary Row operations Solutions to systems of linear equations Definition A solution of a linear system in n unknowns x1 , x2 , . . . , xn is a sequence of n numbers s1 , s2 , . . . , sn for which the substitution x1 = s1 , x2 = s2 , ..., xn = sn makes each equation a true statement. The solution can be written as (s1 , s2 , · · · , sn ) which is called an ordered n-tuple. If n = 2: ordered pair; if n = 3: ordered triple. A linear system is called consistent if it has at least one solution and inconsistent if it has no solution. Dr K Introduction to systems of linear equations Linear Equations Linear systems with two and three unknowns Augmented Matrices Elementary Row operations Solutions to systems of linear equations Definition A solution of a linear system in n unknowns x1 , x2 , . . . , xn is a sequence of n numbers s1 , s2 , . . . , sn for which the substitution x1 = s1 , x2 = s2 , ..., xn = sn makes each equation a true statement. The solution can be written as (s1 , s2 , · · · , sn ) which is called an ordered n-tuple. If n = 2: ordered pair; if n = 3: ordered triple. A linear system is called consistent if it has at least one solution and inconsistent if it has no solution. Dr K Introduction to systems of linear equations Linear Equations Linear systems with two and three unknowns Augmented Matrices Elementary Row operations The solution of Linear systems in two unknowns arise in connection with intersections of lines. For example, consider the linear system a1 x + b1 y = c1 a2 x + b2 y = c2 in which the graphs of the equations are lines in the xy-plane. Each solution (x, y ) of this system corresponds to a point of intersection of the lines, so there are three possibilities. i.e. Dr K Introduction to systems of linear equations Linear Equations Linear systems with two and three unknowns Augmented Matrices Elementary Row operations The solution of Linear systems in two unknowns arise in connection with intersections of lines. For example, consider the linear system a1 x + b1 y = c1 a2 x + b2 y = c2 in which the graphs of the equations are lines in the xy-plane. Each solution (x, y ) of this system corresponds to a point of intersection of the lines, so there are three possibilities. i.e. Dr K Introduction to systems of linear equations Linear Equations Linear systems with two and three unknowns Augmented Matrices Elementary Row operations The solution of Linear systems in two unknowns arise in connection with intersections of lines. For example, consider the linear system a1 x + b1 y = c1 a2 x + b2 y = c2 in which the graphs of the equations are lines in the xy-plane. Each solution (x, y ) of this system corresponds to a point of intersection of the lines, so there are three possibilities. i.e. Dr K Introduction to systems of linear equations Linear Equations Linear systems with two and three unknowns Augmented Matrices Elementary Row operations The lines may be parallel and distinct, in which case there is no intersection and consequently no solution. The lines may intersect at only one point, in which case the system has exactly one solution. The lines may coincide, in which case there are infinitely many points of intersection (the points on the common line) and consequently infinitely many solutions. Dr K Introduction to systems of linear equations Linear Equations Linear systems with two and three unknowns Augmented Matrices Elementary Row operations The lines may be parallel and distinct, in which case there is no intersection and consequently no solution. The lines may intersect at only one point, in which case the system has exactly one solution. The lines may coincide, in which case there are infinitely many points of intersection (the points on the common line) and consequently infinitely many solutions. Dr K Introduction to systems of linear equations Linear Equations Linear systems with two and three unknowns Augmented Matrices Elementary Row operations The lines may be parallel and distinct, in which case there is no intersection and consequently no solution. The lines may intersect at only one point, in which case the system has exactly one solution. The lines may coincide, in which case there are infinitely many points of intersection (the points on the common line) and consequently infinitely many solutions. Dr K Introduction to systems of linear equations Linear Equations Linear systems with two and three unknowns Augmented Matrices Elementary Row operations Dr K Introduction to systems of linear equations Linear Equations Linear systems with two and three unknowns Augmented Matrices Elementary Row operations The same is true for a linear system of three equations in three unknowns a1 x + b1 y + c1 z = d1 a2 x + b2 y + c2 z = d2 a3 x + b3 y + c3 z = d3 in which the graphs of the equations are planes. The solutions of the system, if any, correspond to points where all three planes intersect, so again there are only three possibilities?no solutions,one solution, or infinitely many solutions Dr K Introduction to systems of linear equations Linear Equations Linear systems with two and three unknowns Augmented Matrices Elementary Row operations The same is true for a linear system of three equations in three unknowns a1 x + b1 y + c1 z = d1 a2 x + b2 y + c2 z = d2 a3 x + b3 y + c3 z = d3 in which the graphs of the equations are planes. The solutions of the system, if any, correspond to points where all three planes intersect, so again there are only three possibilities?no solutions,one solution, or infinitely many solutions Dr K Introduction to systems of linear equations Linear Equations Linear systems with two and three unknowns Augmented Matrices Elementary Row operations The same is true for a linear system of three equations in three unknowns a1 x + b1 y + c1 z = d1 a2 x + b2 y + c2 z = d2 a3 x + b3 y + c3 z = d3 in which the graphs of the equations are planes. The solutions of the system, if any, correspond to points where all three planes intersect, so again there are only three possibilities?no solutions,one solution, or infinitely many solutions Dr K Introduction to systems of linear equations Linear Equations Linear systems with two and three unknowns Augmented Matrices Elementary Row operations The same is true for a linear system of three equations in three unknowns a1 x + b1 y + c1 z = d1 a2 x + b2 y + c2 z = d2 a3 x + b3 y + c3 z = d3 in which the graphs of the equations are planes. The solutions of the system, if any, correspond to points where all three planes intersect, so again there are only three possibilities?no solutions,one solution, or infinitely many solutions Dr K Introduction to systems of linear equations Linear Equations Linear systems with two and three unknowns Augmented Matrices Elementary Row operations The same is true for a linear system of three equations in three unknowns a1 x + b1 y + c1 z = d1 a2 x + b2 y + c2 z = d2 a3 x + b3 y + c3 z = d3 in which the graphs of the equations are planes. The solutions of the system, if any, correspond to points where all three planes intersect, so again there are only three possibilities?no solutions,one solution, or infinitely many solutions Dr K Introduction to systems of linear equations Linear Equations Linear systems with two and three unknowns Augmented Matrices Elementary Row operations The same is true for a linear system of three equations in three unknowns a1 x + b1 y + c1 z = d1 a2 x + b2 y + c2 z = d2 a3 x + b3 y + c3 z = d3 in which the graphs of the equations are planes. The solutions of the system, if any, correspond to points where all three planes intersect, so again there are only three possibilities?no solutions,one solution, or infinitely many solutions Dr K Introduction to systems of linear equations Linear Equations Linear systems with two and three unknowns Augmented Matrices Elementary Row operations Dr K Introduction to systems of linear equations Linear Equations Linear systems with two and three unknowns Augmented Matrices Elementary Row operations Example of linear system with one solution Example Solve the linear systems: x +y =1 2x − y = 11 x =4 y = −3 or (x, y ) = (4, −3). Dr K Introduction to systems of linear equations Linear Equations Linear systems with two and three unknowns Augmented Matrices Elementary Row operations Example of linear system with no solution Example Solve the linear systems: x +y =4 3x + 3y = 6 no solution Dr K Introduction to systems of linear equations Linear Equations Linear systems with two and three unknowns Augmented Matrices Elementary Row operations Example of linear system with infinitely many solution Example Solve the linear systems: 4x − 2y = 1 16x − 8y = 4 infinitely many solutions Dr K Introduction to systems of linear equations Linear Equations Linear systems with two and three unknowns Augmented Matrices Elementary Row operations Augmented Matrices Definition The linear system a11 x1 a21 x1 .. . + + a12 x2 a22 x2 .. . + ··· + ··· ··· am1 x1 + am2 x2 + · · · + + a1n xn a2n xn .. . = = b1 b2 .. . + amn xn = bm can be abbreviated to the augmented matrix a11 a12 · · · a1n b1 a21 a22 · · · a2n xn b2 .. .. .. .. . . ··· . . am1 am2 · · · Dr K amn bm Introduction to systems of linear equations Linear Equations Linear systems with two and three unknowns Augmented Matrices Elementary Row operations Augmented matrices Example The system of linear equations x1 + x2 + 2x3 = 9 2x1 + 4x2 − 3x3 = 1 3x1 + 6x2 − 5x3 1 1 has the augmented matrix 2 4 3 6 Dr K =0 2 9 −3 1 −5 0 Introduction to systems of linear equations Linear Equations Linear systems with two and three unknowns Augmented Matrices Elementary Row operations The basic method for solving a linear system is to perform algebraic operations on the system. These operations do not alter the solution set and that produce a succession of increasingly simpler systems, until a point is reached where it can be ascertained whether the system is consistent, and if so, what its solutions are. Typically, the algebraic operations are: Multiply an equation through by a nonzero constant. Interchange two equations. Add a constant times one equation to another. Dr K Introduction to systems of linear equations Linear Equations Linear systems with two and three unknowns Augmented Matrices Elementary Row operations The basic method for solving a linear system is to perform algebraic operations on the system. These operations do not alter the solution set and that produce a succession of increasingly simpler systems, until a point is reached where it can be ascertained whether the system is consistent, and if so, what its solutions are. Typically, the algebraic operations are: Multiply an equation through by a nonzero constant. Interchange two equations. Add a constant times one equation to another. Dr K Introduction to systems of linear equations Linear Equations Linear systems with two and three unknowns Augmented Matrices Elementary Row operations The basic method for solving a linear system is to perform algebraic operations on the system. These operations do not alter the solution set and that produce a succession of increasingly simpler systems, until a point is reached where it can be ascertained whether the system is consistent, and if so, what its solutions are. Typically, the algebraic operations are: Multiply an equation through by a nonzero constant. Interchange two equations. Add a constant times one equation to another. Dr K Introduction to systems of linear equations Linear Equations Linear systems with two and three unknowns Augmented Matrices Elementary Row operations The basic method for solving a linear system is to perform algebraic operations on the system. These operations do not alter the solution set and that produce a succession of increasingly simpler systems, until a point is reached where it can be ascertained whether the system is consistent, and if so, what its solutions are. Typically, the algebraic operations are: Multiply an equation through by a nonzero constant. Interchange two equations. Add a constant times one equation to another. Dr K Introduction to systems of linear equations Linear Equations Linear systems with two and three unknowns Augmented Matrices Elementary Row operations The basic method for solving a linear system is to perform algebraic operations on the system. These operations do not alter the solution set and that produce a succession of increasingly simpler systems, until a point is reached where it can be ascertained whether the system is consistent, and if so, what its solutions are. Typically, the algebraic operations are: Multiply an equation through by a nonzero constant. Interchange two equations. Add a constant times one equation to another. Dr K Introduction to systems of linear equations Linear Equations Linear systems with two and three unknowns Augmented Matrices Elementary Row operations Since the rows (horizontal lines) of an augmented matrix correspond to the equations in the associated system,these three operations correspond to the following operations on the rows of the augmented matrix: Multiply a row through by a nonzero constant. Interchange two rows. Add a constant times one row to another. These are called elementary row operations on a matrix. Dr K Introduction to systems of linear equations Linear Equations Linear systems with two and three unknowns Augmented Matrices Elementary Row operations Since the rows (horizontal lines) of an augmented matrix correspond to the equations in the associated system,these three operations correspond to the following operations on the rows of the augmented matrix: Multiply a row through by a nonzero constant. Interchange two rows. Add a constant times one row to another. These are called elementary row operations on a matrix. Dr K Introduction to systems of linear equations Linear Equations Linear systems with two and three unknowns Augmented Matrices Elementary Row operations Since the rows (horizontal lines) of an augmented matrix correspond to the equations in the associated system,these three operations correspond to the following operations on the rows of the augmented matrix: Multiply a row through by a nonzero constant. Interchange two rows. Add a constant times one row to another. These are called elementary row operations on a matrix. Dr K Introduction to systems of linear equations Linear Equations Linear systems with two and three unknowns Augmented Matrices Elementary Row operations Elementary row operations Definition The following elementary row operations can be performed on the rows (horizontal lines) of an augmented matrix: 1 Multiply a row by a nonzero constant. cRi (Example: −3R2 ) 2 Interchange two rows. Ri ↔ Rj (Example: R1 ↔ R3 ) 3 Add a constant times one row to another row. Ri + kRj (Example: R2 + 4R1 ) Note that: Ri denotes the ith row. The row you are changing must be written first. Dr K Introduction to systems of linear equations Linear Equations Linear systems with two and three unknowns Augmented Matrices Elementary Row operations Example Use row operations to solve the system x + y + 2z = 9 2x + 4y − 3z = 1 3x + 6y − 5z = 0 Dr K Introduction to systems of linear equations Linear Equations Linear systems with two and three unknowns Augmented Matrices Elementary Row operations Solution: Augmented 1 1 2 2 4 −3 3 6 −5 x + y + 2z = 9 2x + 4y − 3z = 1 3x + 6y − 5z = 0 1 1 2 9 R2 −2R1 −→ 0 2 −7 −17 3 6 −5 0 1 1 2 9 −→0 1 − 72 − 17 2 0 3 −11 −27 1 R 2 2 matrix 9 1 0 1 1 2 9 R3 −3R1 −→ 0 2 −7 −17 0 3 −11 −27 1 1 2 R3 −3R2 0 1 − 72 −→ 0 0 − 12 Dr K 9 − 17 2 − 23 Introduction to systems of linear equations Linear Equations Linear systems with two and three unknowns Augmented Matrices Elementary Row operations Solution: Augmented 1 1 2 2 4 −3 3 6 −5 x + y + 2z = 9 2x + 4y − 3z = 1 3x + 6y − 5z = 0 1 1 2 9 R2 −2R1 −→ 0 2 −7 −17 3 6 −5 0 1 1 2 9 −→0 1 − 72 − 17 2 0 3 −11 −27 1 R 2 2 matrix 9 1 0 1 1 2 9 R3 −3R1 −→ 0 2 −7 −17 0 3 −11 −27 1 1 2 R3 −3R2 0 1 − 72 −→ 0 0 − 12 Dr K 9 − 17 2 − 23 Introduction to systems of linear equations Linear Equations Linear systems with two and three unknowns Augmented Matrices Elementary Row operations Solution: Augmented 1 1 2 2 4 −3 3 6 −5 x + y + 2z = 9 2x + 4y − 3z = 1 3x + 6y − 5z = 0 1 1 2 9 R2 −2R1 −→ 0 2 −7 −17 3 6 −5 0 1 1 2 9 −→0 1 − 72 − 17 2 0 3 −11 −27 1 R 2 2 matrix 9 1 0 1 1 2 9 R3 −3R1 −→ 0 2 −7 −17 0 3 −11 −27 1 1 2 R3 −3R2 0 1 − 72 −→ 0 0 − 12 Dr K 9 − 17 2 − 23 Introduction to systems of linear equations Linear Equations Linear systems with two and three unknowns Augmented Matrices Elementary Row operations Solution: Augmented 1 1 2 2 4 −3 3 6 −5 x + y + 2z = 9 2x + 4y − 3z = 1 3x + 6y − 5z = 0 1 1 2 9 R2 −2R1 −→ 0 2 −7 −17 3 6 −5 0 1 1 2 9 −→0 1 − 72 − 17 2 0 3 −11 −27 1 R 2 2 matrix 9 1 0 1 1 2 9 R3 −3R1 −→ 0 2 −7 −17 0 3 −11 −27 1 1 2 R3 −3R2 0 1 − 72 −→ 0 0 − 12 Dr K 9 − 17 2 − 23 Introduction to systems of linear equations Linear Equations Linear systems with two and three unknowns Augmented Matrices Elementary Row operations Solution: Augmented 1 1 2 2 4 −3 3 6 −5 x + y + 2z = 9 2x + 4y − 3z = 1 3x + 6y − 5z = 0 1 1 2 9 R2 −2R1 −→ 0 2 −7 −17 3 6 −5 0 1 1 2 9 −→0 1 − 72 − 17 2 0 3 −11 −27 1 R 2 2 matrix 9 1 0 1 1 2 9 R3 −3R1 −→ 0 2 −7 −17 0 3 −11 −27 1 1 2 R3 −3R2 0 1 − 72 −→ 0 0 − 12 Dr K 9 − 17 2 − 23 Introduction to systems of linear equations Linear Equations Linear systems with two and three unknowns Augmented Matrices Elementary Row operations Solution: Augmented 1 1 2 2 4 −3 3 6 −5 x + y + 2z = 9 2x + 4y − 3z = 1 3x + 6y − 5z = 0 1 1 2 9 R2 −2R1 −→ 0 2 −7 −17 3 6 −5 0 1 1 2 9 −→0 1 − 72 − 17 2 0 3 −11 −27 1 R 2 2 matrix 9 1 0 1 1 2 9 R3 −3R1 −→ 0 2 −7 −17 0 3 −11 −27 1 1 2 R3 −3R2 0 1 − 72 −→ 0 0 − 12 Dr K 9 − 17 2 − 23 Introduction to systems of linear equations Linear Equations Linear systems with two and three unknowns Augmented Matrices Elementary Row operations Example continued 1 1 2 0 1 − 7 2 0 0 − 21 9 − 17 2 − 32 1 0 11 2 R1 −R2 −→ 0 1 − 72 0 0 1 35 2 − 17 2 1 1 2 −2R3 −→ 0 1 − 72 0 0 1 9 − 17 2 3 R1 − 11 R3 2 1 0 0 1 −→ 0 1 0 2 0 0 1 3 R2 + 72 R3 3 The solution x = 1, y = 2, z = 3 is now evident. Dr K Introduction to systems of linear equations Linear Equations Linear systems with two and three unknowns Augmented Matrices Elementary Row operations Example continued 1 1 2 0 1 − 7 2 0 0 − 21 9 − 17 2 − 32 1 0 11 2 R1 −R2 −→ 0 1 − 72 0 0 1 35 2 − 17 2 1 1 2 −2R3 −→ 0 1 − 72 0 0 1 9 − 17 2 3 R1 − 11 R3 2 1 0 0 1 −→ 0 1 0 2 0 0 1 3 R2 + 72 R3 3 The solution x = 1, y = 2, z = 3 is now evident. Dr K Introduction to systems of linear equations Linear Equations Linear systems with two and three unknowns Augmented Matrices Elementary Row operations Example continued 1 1 2 0 1 − 7 2 0 0 − 21 9 − 17 2 − 32 1 0 11 2 R1 −R2 −→ 0 1 − 72 0 0 1 35 2 − 17 2 1 1 2 −2R3 −→ 0 1 − 72 0 0 1 9 − 17 2 3 R1 − 11 R3 2 1 0 0 1 −→ 0 1 0 2 0 0 1 3 R2 + 72 R3 3 The solution x = 1, y = 2, z = 3 is now evident. Dr K Introduction to systems of linear equations Linear Equations Linear systems with two and three unknowns Augmented Matrices Elementary Row operations Example continued 1 1 2 0 1 − 7 2 0 0 − 21 9 − 17 2 − 32 1 0 11 2 R1 −R2 −→ 0 1 − 72 0 0 1 35 2 − 17 2 1 1 2 −2R3 −→ 0 1 − 72 0 0 1 9 − 17 2 3 R1 − 11 R3 2 1 0 0 1 −→ 0 1 0 2 0 0 1 3 R2 + 72 R3 3 The solution x = 1, y = 2, z = 3 is now evident. Dr K Introduction to systems of linear equations Linear Equations Linear systems with two and three unknowns Augmented Matrices Elementary Row operations Example continued 1 1 2 0 1 − 7 2 0 0 − 21 9 − 17 2 − 32 1 0 11 2 R1 −R2 −→ 0 1 − 72 0 0 1 35 2 − 17 2 1 1 2 −2R3 −→ 0 1 − 72 0 0 1 9 − 17 2 3 R1 − 11 R3 2 1 0 0 1 −→ 0 1 0 2 0 0 1 3 R2 + 72 R3 3 The solution x = 1, y = 2, z = 3 is now evident. Dr K Introduction to systems of linear equations Linear Equations Linear systems with two and three unknowns Augmented Matrices Elementary Row operations Example continued 1 1 2 0 1 − 7 2 0 0 − 21 9 − 17 2 − 32 1 0 11 2 R1 −R2 −→ 0 1 − 72 0 0 1 35 2 − 17 2 1 1 2 −2R3 −→ 0 1 − 72 0 0 1 9 − 17 2 3 R1 − 11 R3 2 1 0 0 1 −→ 0 1 0 2 0 0 1 3 R2 + 72 R3 3 The solution x = 1, y = 2, z = 3 is now evident. Dr K Introduction to systems of linear equations Linear Equations Linear systems with two and three unknowns Augmented Matrices Elementary Row operations Example continued 1 1 2 0 1 − 7 2 0 0 − 21 9 − 17 2 − 32 1 0 11 2 R1 −R2 −→ 0 1 − 72 0 0 1 35 2 − 17 2 1 1 2 −2R3 −→ 0 1 − 72 0 0 1 9 − 17 2 3 R1 − 11 R3 2 1 0 0 1 −→ 0 1 0 2 0 0 1 3 R2 + 72 R3 3 The solution x = 1, y = 2, z = 3 is now evident. Dr K Introduction to systems of linear equations Linear Equations Linear systems with two and three unknowns Augmented Matrices Elementary Row operations In conclusion: Do you know these concepts? Linear Equation Consistent linear system Homogenous linear equation Inconsistent linear system System of linear equation Augmented matrix Elementary row operations Solution of a linear system Dr K Introduction to systems of linear equations Linear Equations Linear systems with two and three unknowns Augmented Matrices Elementary Row operations Can you? Determine whether a given equation is linear. Determine whether a given n-tuple is a solution of a linear system Find the augmented matrix of a linear system Find the linear system corresponding to a given augmented matrix Perform elementary row operations on a linear system and on its corresponding augmented matrix Determine whether a linear system is consistent or inconsistent Find the solution to a consistent linear system Dr K Introduction to systems of linear equations Linear Equations Linear systems with two and three unknowns Augmented Matrices Elementary Row operations Exercise: Solve the following linear systems (a) x + 3y + z = 5 2x + 7y + 2z = 4 x + 2y + 2z = 3 (b) 3y − z = 4 x + 3y + z = 1 x + 2z = 4 (c) 2x + 4y − 6z = 2 −x − 2y + 3z = −1 x +y +z = 3 (a) (x, y , z) = (31, −6, −8) (b) Inconsistent. (c)(x, y , z) = (5, −2, 0) + t(−5, 4, 1) Dr K Introduction to systems of linear equations