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
- Three sheets of graph paper
- A ruler
- A pencil
Theory
The lines corresponding to each of the equations given in a system of linear equations are drawn on a graph paper. Now,
- if the two lines intersect at a point then the system is consistent and has a unique solution.
- if the two lines are coincident then the system is consistent and has infinitely many solutions.
- if the two lines are parallel to each other then the system is inconsistent and has no solution.
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
- The first system of equations is represented by intersecting lines, which shows that the system is consistent and has a unique solution, i.e., x = -1, y = 2 (see the first observation table).
- The second system of equations is represented by coincident lines, which shows that the system is consistent and has infinitely many solutions (see the second observation table).
- The third system of equations is represented by parallel lines, which shows that the system is inconsistent and has no solution (see the third observation table).
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)
a₁ b₁ c₁
---- = ---- ≠ ----
a₂ b₂ c₂
3. For 3rd pair of equations (coincident lines)
a₁ b₁ c₁
---- = ---- = ----
a₂ b₂ c₂
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