OBJECTIVE To use the graphical method to obtain the conditions of consistency and hence to solve a given system of linear equations in two variables Materials Required
Theory
Procedure We shall consider a pair of linear equations in two variables of the type a1x +b1y = c1 a2x +b2y = c2 Step 1: Let the first system of linear equations be x + 2y = 3 … (i) 4x + 3y = 2 … (ii) Step 2: From equation (i), we have y= ½(3 – x).Find the values of y for two different values of x as shown below. Similarly, from equation (ii), we have y=1/3( 2 – 4x). Then Step 3: Draw a line representing the equation x+2y = 3 on graph paper I by plotting the points (1,1) and (3,0), and joining them. Similarly, draw a line representing the equation 4x + 3y = 2 by plotting the points (-1, 2) and (2, -2), and joining them.
Step 4: Record your observations in the first observation table. Step 5: Consider a second system of linear equations: x – 2y = 3 … (iii) -2x + 4y = -6 … (iv) Step 6: From equation (iii), we get From equation (iv), we get Draw lines on graph paper II using these points and record your observations in the second observation table. Step 7: Consider a third system of linear equations: 2x – 3y = 5 …(v) -4x + 6y = 3 … (vi) Step 8: From equation (v), we get From equation (vi), we get Draw lines on graph paper III using these points and record your observations in the third observation table. Observations I. For the first system of equations II. For the second system of equations III. For the third system of equations Conclusions
Remarks: The teacher must provide the students with additional problems for practice of each of the three types of systems of equations. Math Labs with ActivityMath LabsScience Practical SkillsScience Labs
Objective To verify the conditions for consistency of a system of linear equations in two variables by graphical representation. Prerequisite Knowledge 1. Plotting of points on a graph paper. 2. Condition of consistency of lines parallel, intersecting, coincident, Materials Required Graph papers, fevicol, geometry box, cardboard. Procedure Consider the three pairs of linear equations 1st pair: 2x-5y+4=0, 2x+y-8 = 0 2nd pair: 4x + 6y = 24, 2x + 3y =6 3rd pair: x-2y=5, 3x-6y=15 1. Take the 1st pair of linear equations in two variables, e.g., 2x – 5y +4=0, 2x +y- 8 = 0. 2. Obtain a table of at least three such pairs (x, y) which satisfy the given equations. 3. Plot the points of two equations on the graph paper as shown below. 4. Observe whether the lines are intersecting, parallel or coincident. Write the values in observation table. Also, check for: a₁ b₁ c₁ ---- = ---- = ---- a₂ b₂ c₂ 5. Take the second pair of linear equations in two variables 6. Repeat the steps 3 and 4 and draw linear graph as shown. 7. Take the third pair of linear equations in two variables,i.e. x-2y=5, 3x-6y=15. Plot XY values. 8. Repeat steps 3 and 4 and plot the graph (as shown) Observation: Following are the observations:
1. In 1st pair, for intersecting lines 2. For second pair (parallel lines) 3. For 3rd pair of equations (coincident lines)
Result The conditions for consistency of a system of linear equations in two variables is verified. Project Activity - What's so special about Ramanujam's Magic Square
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