Question

Solve each linear equation and fill each blank square with correct answer. Note :- Number must appear only once in each row, column and block. (i) X + 2 = 7 (ii) 2x = 14 (iii) 3x – 1 = 15 + x (iv) − = 2

Answer 1

Answer:

1.1 Solutions and elementary operations

Practical problems in many fields of study—such as biology, business, chemistry, computer science, economics, electronics, engineering, physics and the social sciences—can often be reduced to solving a system of linear equations. Linear algebra arose from attempts to find systematic methods for solving these systems, so it is natural to begin this book by studying linear equations.

If a, b, and c are real numbers, the graph of an equation of the form

\begin{equation*} ax + by = c \end{equation*}

is a straight line (if a and b are not both zero), so such an equation is called a linear equation in the variables x and y. However, it is often convenient to write the variables as x_1, x_2, \dots, x_n, particularly when more than two variables are involved. An equation of the form

\begin{equation*} a_1x_1 + a_2x_2 + \dots + a_nx_n = b \end{equation*}

is called a linear equation in the n variables x_1, x_2, \dots, x_n. Here a_1, a_2, \dots, a_n denote real numbers (called the coefficients of x_1, x_2, \dots, x_n, respectively) and b is also a number (called the constant term of the equation). A finite collection of linear equations in the variables x_1, x_2, \dots, x_n is called a system of linear equations in these variables. Hence,

\begin{equation*} 2x_1 - 3x_2 + 5x_3 = 7 \end{equation*}

is a linear equation; the coefficients of x_1, x_2, and x_3 are 2, -3, and 5, and the constant term is 7. Note that each variable in a linear equation occurs to the first power only.

Given a linear equation a_1x_1 + a_2x_2 + \dots + a_nx_n = b, a sequence s_1, s_2, \dots, s_n of n numbers is called a solution to the equation if

\begin{equation*} a_1s_1 + a_2s_2 + \dots + a_ns_n = b \end{equation*}

that is, if the equation is satisfied when the substitutions x_1 = s_1, x_2 = s_2, \dots, x_n = s_n are made. A sequence of numbers is called a solution to a system of equations if it is a solution to every equation in the system.