1. Algebraic Method of Solving a Pair of Linear Equations.
In mathematics, a system of linear equations (or linear system) is a collection of two or more linear equations involving the same set of variables. The word "system" indicates that the equations are to be considered collectively, rather than individually. A solution to this system of equations is the pair of values of the unknown variables x and y that satisfy both equations.
Algebraic Method of solving a pair of Linear Equations in 2-Variables
(1) Substitution method
Re-write one of the equations in any one of the variables (in terms of the other variable).
Substitute this in the second equation and solve for the second variable.
(2) Elimination method
Multiply both the equations with suitable non-zero constants, so that the coefficients of one of the
variables in both equations become equal.
Subtract one equation from another, to eliminate the variable with equal coefficients.
Solve for the remaining variable.
(3) Cross – multiplication method
Arrange the variables x and y, their coefficients a1, a2, b1, b2, and the constants c1, c2, then solve for x
and y.
2. Graphical method of solution
For a system of linear equations involving two variables (x and y), each linear equation can be represented as a line on the coordinate axes. Since a solution to the linear system must satisfy all of the equations, the solution set will be the intersection of these lines, which is either a line, a single point, or the empty set.
General Form of a System of Linear Equations
A system of linear equations (or linear system) is a collection of two or more linear equations involving the same set of variables. For example,
3x + 2y = 1 – (1)
2x – 2y = -6
is a system of two equations in the two variables x, y.
A solution to a linear system is a set of values, one for each variable, such that both the equations are simultaneously satisfied. A solution to the system above is given by
x = -1
y = 2
since it satisfies both the equations.
The general form of a system of linear equations in two variables is written as
a1x + b1y + c1 = 0
a2x + b2y + c2 = 0 , where a1, a2, b1, b2, c1, c2 are all real numbers, a12 + b12 ≠ 0 and a22 + b22 ≠ 0
For example, in (1),
a1 = 3, b1 = 2, c1 = -1
Graphical Method of Solving a System of Linear Equations
A linear equation in two variables when plotted on a graph defines a line. So, this means when a pair of linear equations is plotted, two lines are defined. Now, there are two lines in a plane These lines can:
- intersect each other,
- be parallel to each other, or
- coincide with each other.
The point(s) where the two lines intersect will give the solutions of the pair of linear equations, graphically.
Condition 1: Intersecting Lines
If spa id="MathJax-Element-1-Frame" class="MathJax" style="box-sizing: border-box; margin: 0px; padding: 0px; word-break: break-word; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 1em; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;" tabindex="0" data-mathml="a1a2">a1a2a1a2 ≠ span i="MathJax-Element-2-Frame" class="MathJax" style="box-sizing: border-box; margin: 0px; padding: 0px; word-break: break-word; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 1em; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;" tabindex="0" data-mathml="b1b2">b1b2b1b2, then the pair of linear equations a1x + b1y + c1 = 0, a2x + b2y + c2 = 0 has a unique solution.
Condition 2: Coincident Lines
If spanid="MathJax-Element-3-Frame" class="MathJax" style="box-sizing: border-box; margin: 0px; padding: 0px; word-break: break-word; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 1em; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;" tabindex="0" data-mathml="a1a2">a1a2a1a2 = spanid="MathJax-Element-4-Frame" class="MathJax" style="box-sizing: border-box; margin: 0px; padding: 0px; word-break: break-word; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 1em; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;" tabindex="0" data-mathml="b1b2">b1b2b1b2 = spanid="MathJax-Element-5-Frame" class="MathJax" style="box-sizing: border-box; margin: 0px; padding: 0px; word-break: break-word; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 1em; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;" tabindex="0" data-mathml="c1c2">c1c2c1c2, then the pair of linear equations a1x + b1y + c1 = 0, a2x + b2y + c2 = 0 has infinite solutions.
A pair of linear equations, which has unique or infinite solutions is said to be a consistent pair of linear equations.
Condition 3: Parallel Lines
If span i="MathJax-Element-6-Frame" class="MathJax" style="box-sizing: border-box; margin: 0px; padding: 0px; word-break: break-word; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 1em; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;" tabindex="0" data-mathml="a1a2">a1a2a1a2 = spanid="MathJax-Element-7-Frame" class="MathJax" style="box-sizing: border-box; margin: 0px; padding: 0px; word-break: break-word; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 1em; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;" tabindex="0" data-mathml="b1b2">b1b2b1b2 ≠ span id=MathJax-Element-8-Frame" class="MathJax" style="box-sizing: border-box; margin: 0px; padding: 0px; word-break: break-word; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 1em; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;" tabindex="0" data-mathml="c1c2">c1c2c1c2, then a pair of linear equations a1x + b1y + c1 = 0, a2x + b2y + c2 = 0 has no solution.
A pair of linear equations which has no solution is said to be an inconsistent pair of linear equations.
3. Equations Reducible to Pair of Linear Equations in Two Variables
There are several cases in which the problem can be mathematically represented by two equations that are not linear. However, by making suitable substitutions, these equations can be re-written as linear equations.
Equations Reducible to a Pair of Linear Equations
Some equations, which are not linear, can be reduced to linear equations by making necessary substitutions, as shown in the example below:
Example:
1. Solve the following pair of equations by reducing them to pairs of linear equations:
span i="MathJax-Element-9-Frame" class="MathJax" style="box-sizing: border-box; margin: 0px; padding: 0px; word-break: break-word; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 1em; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;" tabindex="0" data-mathml="2x">2x2x + spanid="MathJax-Element-10-Frame" class="MathJax" style="box-sizing: border-box; margin: 0px; padding: 0px; word-break: break-word; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 1em; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;" tabindex="0" data-mathml="3y">3y3y = 13
span i="MathJax-Element-11-Frame" class="MathJax" style="box-sizing: border-box; margin: 0px; padding: 0px; word-break: break-word; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 1em; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;" tabindex="0" data-mathml="5x">5x5x + spanid="MathJax-Element-12-Frame" class="MathJax" style="box-sizing: border-box; margin: 0px; padding: 0px; word-break: break-word; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 1em; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;" tabindex="0" data-mathml="4y">4y4y= -2
Sol:
Substituting span id="Mathax-Element-13-Frame" class="MathJax" style="box-sizing: border-box; margin: 0px; padding: 0px; word-break: break-word; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 1em; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;" tabindex="0" data-mathml="1x">1x1x = p and span id="MthJax-Element-14-Frame" class="MathJax" style="box-sizing: border-box; margin: 0px; padding: 0px; word-break: break-word; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 1em; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;" tabindex="0" data-mathml="1y">1y1y = q in the given equations, we get
2p + 3q = 13 ……………………. (i)
5p – 4q = – 2 ……………………..(ii)
Multiplying eq. (i) by 4 and eq. (2) by 3, we get
8p + 12q = 52 ………………….(iii)
15p – 12q = -6 …………………(iv)
Substituting this value of p in equation (i), we get
2 × 2 + 3q = 13
3q = 9
q = 3
Therefore, span id="MahJax-Element-15-Frame" class="MathJax" style="box-sizing: border-box; margin: 0px; padding: 0px; word-break: break-word; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 1em; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;" tabindex="0" data-mathml="1x">1x1x= 2 and span id=MathJax-Element-16-Frame" class="MathJax" style="box-sizing: border-box; margin: 0px; padding: 0px; word-break: break-word; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 1em; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;" tabindex="0" data-mathml="1y">1y1y= 3
Hence, the solution of the given pair of equations is
x = span id="MathJax-Element-17-Frme" class="MathJax" style="box-sizing: border-box; margin: 0px; padding: 0px; word-break: break-word; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 1em; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;" tabindex="0" data-mathml="12">1212 and y = span id="MthJax-Element-18-Frame" class="MathJax" style="box-sizing: border-box; margin: 0px; padding: 0px; word-break: break-word; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 1em; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;" tabindex="0" data-mathml="13">13