**Math Labs with Activity – Solve the System of Linear Equations**

**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

a_{1}x +b_{1}y = c_{1}

a_{2}x +b_{2}y = c_{2}

**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.

x | 1 | 3 |

y | 1 | 0 |

Similarly, from equation (ii), we have

y=1/3( 2 – 4x).

Then

x | -1 | 2 |

y | 2 | -2 |

**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

x | 3 | 1 |

y | 0 | -l |

From equation (iv), we get

x | -3 | -1 |

y | -3 | -2 |

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

x | 1 | 4 |

y | -1 | 1 |

From equation (vi), we get

x | 0 | 3 |

y | ½ | 5/2 |

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.

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