1. GRAPHS AND LINEAR EQUATIONS

2. LINEAR EQUATION A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and (the first power of) a single variable.algebraic equationtermconstantvariable Linear equations can have one or more variables. Linear equations occur with great regularity in applied mathematics.applied mathematics

3. Linear equations in two variables A common form of a linear equation in the two variables x and y is  Y = mx + b where m and b designate constants. The origin of the name "linear" comes from the fact that the set of solutions of such an equation forms a straight line in the plane. In this particular equation, the constant m determines the slope or gradient of that line, and the constant term "b" determines the point at which the line crosses the y-axis, otherwise known as the y-intercept.straight line slopeconstant term Since terms of linear equations cannot contain products of distinct or equal variables, nor any power (other than 1) or other function of a variable, equations involving terms such as xy, x2, y1/3, and sin(x) are nonlinear.nonlinear

4. Slope–intercept form Y = mx + b  where m is the slope of the line and b is the y- intercept, which is the y-coordinate of the point where the line crosses the y axis. This can be seen by letting x = 0, which immediately gives y = b. Vertical lines, having undefined slope, cannot be represented by this form.

5. Point–slope form Y – y1 = m(x – x1)  where m is the slope of the line and (x1,y1) is any point on the line.  The point-slope form expresses the fact that the difference in the y coordinate between two points on a line (that is, y − y1) is proportional to the difference in the x coordinate (that is, x − x1). The proportionality constant is m (the slope of the line)

6. How to Graph Linear Equations How to Graph Linear Equation Of y = mx + c Method Step1:- Choose the convenient values of x and find the corresponding values of y Step2:- Prepare a table for different pairs of values of x and y Step3:- Draw the axes on a graph paper and chose a suitable sale. Step4:- Plot the ordered pairs fro the above table on the graph paper Step5:- Join these points by a straight line This straight line is the graph of y = mx + c

7. Example : How to Graph Linear Equations of the line y = 2x + 3. Write down its (i) y-intercept (ii) slope.

8. Solution We have x = 1 → y = (2 × 1 = 3) = 5 x = –2 → y = {2 × (–2) + 3} = – 1 Thus, we have the following table x 1 – 2 y 5 – 1 Plot the Points A(1, 5) and B(–2, –1) on a graph paper. Join AB and produce it. Then, AB is the required graph of the line y = 2x + 3 clearly, (i) y-intercept = 3 (ii) slope = 2

10. Linear equation in One Variable y = 5 Consider the equation y - 5 = 0 y - 5 = 0 y = 5 Since this equation is independent of x, for all values of x, y = 5. Hence the graph of y = 5 is a line parallel to the x-axis at y = 5.

12. Example on Linear Equation in One Variable Graph Question : Plot the graph of 4x+y=4 X = (4 – y)/4 Put y = -4, x = 4-(-4)/4 = 8/4 x=2 Put y = 8, x = 4-8/4 x=-1 Put y = -8, x = 4 + 8 / 4 x=3