Introduction Sometimes graphing a single linear equation is all it takes to solve a mathematical problem. This is often the case when a problem involves two variables.
A square array of quantities, called elements, symbolizing the sum of certain products of these elements. The symbol denotes a determinant of order n. It is an abbreviation for the algebraic sum of all possible products where each product of n factors contains one and only one element from each row and one and only one element from each column.
There will be n! Each product has a plus or minus sign attached to it according as the column indices form an even or odd permutation when the row indices are in natural order i.
For example, the term a13a21a34a42 of the expansion of a determinant of order four has the column indices in order 3,1,4,2.
This term should have a negative sign attached, since three successive interchanges will change the column indices to 1,3,4,21,3,2,4 and 1,2,3,4the last being in natural order.
Determinants of the second order. The value of the determinant of order two is given by The solution of a system of two linear equations in two unknowns given in terms of determinants. The solution of the linear system is providing the equations are consistent and independent. The equations are consistent and independent if and only if Note that the denominator determinants are the same for both x and y and consist of the coefficients of the variables x and y arranged exactly as they appear in the left members of 1.
This denominator determinant is called the determinant of the coefficients or the determinant of the system. The numerator determinants are constructed from this denominator determinant by replacing the column containing the coefficients of the variable being solved for by the column of constants, c1 and c2, from the right side of 1 i.
Determinants of the third order. The value of the determinant is given by This sum can be remembered by the device shown in Fig. The solution of a system of three linear equations in three unknowns given in terms of determinants.
The solution of the linear system is given by providing the equations are consistent and independent. The equations are consistent and independent if and only if Determinants are most easily evaluated by a technique employing the following properties of determinants.
If all the elements of a column or row are zero, the value of the determinant is zero. If each of the elements in a row or column of a determinant is multiplied by the same number p, the value of the determinant is multiplied by p.
If two columns or rows are identical, the value of the determinant is zero. Interchanging any two rows or columns reverses the sign of the determinant.
The value of a determinant is unaltered when all the corresponding rows and columns are interchanged. Thus any theorem proved true for rows holds for columns, and conversely. If each element of a row or column of a determinant is expressed as the sum of two or more terms, the determinant can be expressed as the sum of two or more determinants.
If to each element of a row or column of a determinant is added m times the corresponding element of another row or column the value of the determinant is not changed. Minor of an element in a determinant.
The determinant, of next lower order, obtained by striking out the row and column in which the element lies. Cofactor of an element in a determinant.
Denote the minor of element aij of the i-th row and j-th column of a determinant A by Mij.Here is a system with no solutions, x+y=8-x-y=-7 Proof: Suppose there exists a solution pair, call it x,y. Then the equations are satisfied i.e. x+y=8 and -x-y= Add the equations to get 0=1. But this can't be true, so there is no solution pair to the system by contradiction.
That is, the resulting system has the same solution set as the original system. Our strategy in solving linear systems, therefore, is to take an augmented matrix for a system and carry it by means of elementary row operations to an equivalent augmented matrix from which the solutions of the system are easily obtained.
kcc1 Count to by ones and by tens. kcc2 Count forward beginning from a given number within the known sequence (instead of having to begin at 1). kcc3 Write numbers from 0 to Represent a number of objects with a written numeral (with 0 representing a count of no objects).
kcc4a When counting objects, say the number names in the standard order, pairing each object with one and only. The solution to a system of linear equations is the ordered pair (or pairs) that satisfies all equations in the system. The solution is the ordered pair(s) common to all lines in the system when the lines are graphed.
Lines that cross at a point (or points) are defined as a consistent system of equations.
The place(s) where they cross are the. The process of solving a linear system of equations by adding or subtracting equations to remove one of the variables. Substitution The process of solving a linear system of equations when one variable be replaced with an equation representing that variable in the other equation of a linear system.
We'll make a linear system (a system of linear equations) whose only solution in (4, -3). First note that there are several (or many) ways to do this. We'll look at two ways: Standard Form Linear Equations A linear equation can be written in several forms.