Business Mathematics Chapter 2 Linear Equations Definition Examples Definition Linear equations are first degree equations Each variable in the equation is raised to the first power A linear equation involving two variables x and y has the standard form ax by c where a b and c are constants and a and b cannot both equal zero Note The presence of terms having power other than 1 or product of variables e g xy would exclude an equation from being linear Name of the variables may be different from x and y 1 3 4 7 is linear equation where 3 4 7 2 5 is non liner equation as power of is not 1 Given a linear equation ax by c the solution set for the equation 2 1 is the set of all ordered pairs x y which satisfy the equation Solution set of an equation For any linear equation S consists of an infinite number of elements Method 1 Assume a value of one variable 2 Substitute this into the equation 3 Solve for the other variable Given 2 4 16 determine the pair of Example values which satisfy the equation when x 2 Solution Put x 2 in given equation gives us 4y 16 4 i e y 3 So the pair 2 3 is a pair of values satisfying the given equation Linear equation with n variables Definition a1 x1 a2 x2 an xn b The solution set S of a linear equation with n variables as Definition A linear equation involving n variables x1 x2 xn has the general form where 1 2 are non zero defined above is the collection of n tuples 1 2 such that 1 1 1 Given an equation 4 1 2 2 6 3 0 what values satisfy the equation when 1 2 and 3 1 Solution Put the given values of 1 and 3 in the above equation gives 2 7 Thus 2 7 1 is a solution of the As in the case of two variables there are infinitely many values in the solution set Example above equation Graphing two variable equations A linear equation involving two variables graphs as a straight line in two dimensions Method 1 Set one variable equal to zero 2 Solve for the value of other variable 3 Set second variable equal to zero 4 Solve for the value of first variable 5 The ordered pairs 0 y and x 0 lie on the line 6 Connect these points and extend the line in both directions Graph the linear equation 2 4 16 Example x intercept y intercept The x intercept of an equation is the point where the graph of the equation crosses the x axis i e y 0 The y intercept of an equation is the point where the graph of the equation crosses the y axis i e x 0 Note Equations of the form x k has no y intercept and equations of the form y k has no x intercept Slope Any straight line with the exception of vertical lines can be characterized by its slope Slope represents the inclination of a line or equivalently it shows the rate at which the line raises and fall or how steep the line is Explanation The slope of a line may be positive negative zero or undefined The line with slope 1 Positive then the line rises from left to right 2 Negative then the line falls from left to right 3 Zero then the line is horizontal line 4 Undefined if the line is vertical line Note The sign of the slope represents whether the line falling or raising Its magnitude shows the steepness of the line Two point formula slope Given any two point which lie on a non vertical straight line the slope can be computed as the ratio of change in the value of y to the change in the value of x Slope change in the value of y change in the vale of x Slope 2 1 2 1 Solution Here we have 1 1 2 3 and 2 2 1 9 so Two point formula mathematically The slope m of a straight line connecting two points x1 y 1 and x 2 y 2 is given by the formula Compute the slope of the line segment connecting the two points 2 3 and 1 9 using the above formula we get Slope 9 3 Example We can write it as Slope Intercept form The above equation is called the slope intercept form Generally it is written as 1 2 4 Note Along any straight line the slope is constant The line connecting any two points will have the same slope Consider the general form of two variable equation as slope y intercept 2 4 3 and find the slope and Solution Rewriting the above we get 6 0 Thus slope Rewrite the equation intercept Example 13 is 13 6 and y intercept is zero Slope and Intercept Determining the equation of a straight line have m 5 k 15 We can write down 5 15 as an equation of line i e 5 15 This is the easiest situation to find an equation of line if slope of a line is 5 and y intercept is 0 15 then we Slope and one point If we are given the slope of a line and some point that lies on the line we can substitute the know slope m and Example Given a non vertical straight line with slope m and containing the point x1 y1 the slope of the line connecting x1 y1 with any other point x y is given by Find the equation of line having slope m 2 and passing through the point 2 8 the values in the above equation yields coordinates of the given point into and solve for 1 1 Rearranging gives 1 1 Solution Here 1 1 2 8 and 2 so putting 8 2 2 2 12 Two lines are parallel if they have the same slope i e 1 2 negative reciprocal of each other i e 1 2 1 parallel to the line 8 4 20 Solution From the given equation we have 2 5 Let 1 2 is the slope of the line Then slope of the required line is same i e 2 2 Thus required 4 2 2 2 8 Example Find an equation of line through the point 2 4 and Two lines are perpendicular if their slopes are equal to the equation of line is Parallel and perpendicular lines Chapter 3 Systems of Linear Equations Definition A System of Equations is a set consisting of more than one equation In other words it has dimensions In solving Dimension One way to characterize a system of equations is by its dimensions If system of equations has m equations and n variables then the system is called an m by n system systems of equations we are interested in identifying values of the variables that satisfy all equations in the system simultaneously Definition The solution set for a system of linear equations may be a Null set …