# Number of Solutions to a System of Equations Algebraically

A statement that two mathematical expressions of one or more variables are identical is called an *equation*. *Linear equations* are those in which the powers of all the variables concerned are equal. A linear equation’s degree is always one. A solution of the simultaneous pair of linear equations is a pair of values of variables “x” and “y” that satisfy all equations in the specified set of equations.

### Pair of Linear Equations in Two Variables

An equation that can be put in the form **ax + by + c = 0**, where a, b and c are real numbers, and a and b are not both zero, is called a * linear equation in two variables* x and y. (or this condition should be satisfied where a and b are not both zero by a

^{2}+ b

^{2 }≠ 0).

e.g.:Consider a pair of linear equations in two variables be 3x + 2y = 6,Substitute x = 2 and y = 0 in the left-hand side (LHS) as:

⇒ LHS = 3(2) + 2(0)

= 6 + 0

= 6 = RHS

Therefore, x = 2 and y = 0 is a solution of the equation 2x + 3y = 6.Now, if x = 1 and y = 1 is substituted in the equation 2x + 3y = 6, then:

LHS = 3(1) + 2(1)

= 3 + 2

= 5 ≠ RHS

Therefore, x = 1 and y = 1 is not a solution of the equation.Algebraically, this indicates that the point (2, 0) lies on the line representing the equation 3x + 2y = 6, and the point (1, 1) does not lie on it.

Thus, every solution that satisfies the equation is a point on the line representing it.

Two linear equations like this, having two variables x and y. Equations like these are called a **pair of linear equations in two variables**.

Algebraically, the general form for a pair of linear equations in two variables x and y is:

**a _{1} x + b_{1} y + c_{1 }= 0 **and

**a _{2} x + b_{2} y + c_{2} = 0**

where a_{1}, b_{1}, c_{1}, a_{2}, b_{2}, c_{2} are all real numbers (**∈ R**) and a_{1}^{2} + b_{1}^{2} ≠ 0, a_{2}^{2}+ b_{2} ^{2} ≠ 0.

### Different cases possible for the pair of linear equations in two variables

There are three different cases for three different types of lines: intersecting, parallel and coincident lines to determine the pair of linear equations in two variables.

- The lines may
intersecteach other at a single point.As a result, the pair of equations has

a unique solution(consistent pair of equations).

e.g.:2x – 4y = 0 and 6x + 8y – 40 = 0

- The lines may be
parallelto each other.As a result, the equations have

no solution(inconsistent pair of equations).

e.g.:2x + 4y – 8 = 0 and 4x + 8y – 24 = 0

- The lines may be
coincident.As a result, the equations have

infinitely many solutions(dependent or consistent pair of equations)

e.g.:4x + 6y – 18 = 0 and 8x + 12y – 36 = 0

Now, Lets assume the pair of equations: a_{1}x + b_{1}y + c_{1} = 0 and a_{2}x + b_{2}y + c_{2} = 0 now their graphical and algebraic interpretation corresponding to the comparing ratios is given as:

Comparing ratios | Graphical Representation | Algebraic Interpretation |
---|---|---|

Intersecting Lines | Exactly one Solution (unique) | |

Coincident Lines | Infinitely many solutions | |

Parallel Lines | No solution |

### Sample Problems

**Problem 1: Find out the graphical representation and specify the number of solutions of the following pairs of linear equations: 8x – 4y + 10 = 0 and 4x – 2y + 9 = 0.**

**Solution:**

For the given pair of linear equations:

a

_{1}= 8, b_{1}= -4, c_{1}= 10 anda

_{2}= 4, b_{2}= -2, c_{2}= 9Therefore,

a

_{1}/ a_{2}= 8 / 4 = 2b

_{1}/ b_{2}= -4 / -2 = 2 andc

_{1}/ c_{2}= 10 / 9This implies that:

Hence, the given pairs of linear equations have

no solutionand the lines are parallel and never intersect each other.

**Problem 2: Determine the number of solutions of the following pairs of linear equations: 6x + 2y = 4 and 7x – 3y = 13.**

For the given pair of linear equations:

a

_{1}= 6, b_{1}= 2, c_{1}= -4 anda

_{2}= 7, b_{2}= -3, c_{2}= -13Therefore,

a

_{1}/ a_{2}= 6 / 7b

_{1}/ b_{2}= 2 / -3 andc

_{1}/ c_{2}= -4 / -13This implies that:

Hence, the given pairs of linear equations have

a unique solutionand the lines intersect each other at exactly one point.

**Problem 3: Determine the graphical representation and the number of solutions of the following pairs of linear equations: 6x – 5y = 11; – 12x +10y = –22.**

For the given pair of linear equations:

a

_{1}= 6, b_{1}= -5, c_{1}= -11 anda

_{2}= -12, b_{2}= 10, c_{2}= 22Therefore,

a

_{1}/ a_{2}= 6 / -12 = -1 / 2b

_{1}/ b_{2}= -5 / 10 = -1 / 2 andc

_{1}/ c_{2}= -11 / 22This implies that:

Hence, the given pairs of linear equations have

infinite many solutionsand the lines are coincident.

**Problem 4: Make the pair of linear equations in the following word problems, and find their solutions graphically.**

**There are 20 students in a class. If the number of boys is 6 more than the number of girls, find the number of boys and girls in the class.**

**Solution:**

Consider, the number of girls be x and the number of boys be y.

Therefore, according to the given conditions:

x + y = 20 ……(1)

x – y = 6 ……(2)

In order, to construct the graph the solutions of the given equation is needed to be determined.

For equation (1): x + y = 20, the solutions are:

x y 0 20 20 0 For equation (2): x – y = 6, So, the solutions are:

x y 0 -6 6 0 Thus, plotting the above points for equation (1) and (2) as:

Now, from the graph it can be concluded that the given lines intersect each other at point (13, 7).

Hence, the number of girls are 7 and number of boys are 13 in a class.

**Problem 5: Determine other linear equation in two variables such that it should form:**

**(i) intersecting lines with the line 6x + 7y – 8 = 0.**

**(ii) parallel lines with the line 4x-5y-8=0.**

**Solution:**

(i)In order to make a pair of lines intersect, they must satisfy the given conditions:

On rearranging the above expression as:

Therefore, in the equation the ratio should not be 6/7.

So, another equation can be 5x – 9y + 9 = 0

where the ratio is 5 / -9 and

(ii)In order to make a pair of lines parallel, they should satisfy the given conditions:

On rearranging the above expression as:

Therefore, the required equation a

_{2 }/ b_{2}should be in ratio of 4 / -5 and b_{2}/ c_{2}should not be equal to -5 / -8.So, another equation can be 8x – 10y + 9 = 0

where the ratio a

_{2}/ b_{2}is 8/-10 = -8/10 and

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