Basics Revisited
Equation
An equation is a statement that two mathematical expressions having one or more variables are equal.
Linear Equation
Equations in which the powers of all the variables involved are one are called linear equations. The degree of a linear equation is always one.
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General form of a Linear Equation in Two Variables
The general form of a linear equation in two variables is ax + by + c = 0, where a and b cannot be zero simultaneously.
Representing linear equations for a word problem
To represent a word problem as a linear equation
- Identify unknown quantities and denote them by variables.
- Represent the relationships between quantities in a mathematical form, replacing the unknowns with variables.
Solution of a Linear Equation in 2 variables
The solution of a linear equation in two variables is a pair of values, one for x and the other for y, which makes the two sides of the equation equal.
Eg: If 2x+y=4, then (0,4) is one of its solutions as it satisfies the equation. A linear equation in two variables has infinitely many solutions.
Geometrical Representation of a Linear Equation
Geometrically, a linear equation in two variables can be represented as a straight line.
2x – y + 1 = 0
⇒ y = 2x + 1
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Plotting a Straight Line
The graph of a linear equation in two variables is a straight line. We plot the straight line as follows:
Any additional points plotted in this manner will lie on the same line.
All about Lines
General form of a pair of linear equations in 2 variables
A pair of linear equations in two variables can be represented as follows
The coefficients of x and y cannot be zero simultaneously for an equation.
Nature of 2 straight lines in a plane
For a pair of straight lines on a plane, there are three possibilities
i) They intersect at exactly one point
ii) They are parallel
iii) They are coincident
Graphical Solution
Representing pair of LE in 2 variables graphically
Graphically, a pair of linear equations in two variables can be represented by a pair of straight lines.
Graphical method of finding solution of a pair of Linear Equations
Graphical Method of finding the solution to a pair of linear equations is as follows:
- Plot both the equations (two straight lines)
- Find the point of intersection of the lines.
The point of intersection is the solution.
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Comparing the ratios of coefficients of a Linear Equation
Algebraic Solution
Finding solution for consistent pair of Linear Equations
The solution of a pair of linear equations is of the form (x, y) which satisfies both the equations simultaneously. Solution for a consistent pair of linear equations can be found out using
i) Elimination method
ii) Substitution Method
iii) Cross-multiplication method
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Substitution Method of finding solution of a pair of Linear Equations
Substitution method:
y – 2x = 1
x + 2y = 12
(i) express one variable in terms of the other using one of the equations. In this case, y = 2x + 1.
(ii) substitute for this variable (y) in the second equation to get a linear equation in one variable, x. x + 2 × (2x + 1) = 12
⇒ 5 x + 2 = 12
(iii) Solve the linear equation in one variable to find the value of that variable.
5 x + 2 = 12
⇒ x = 2
(iv) Substitute this value in one of the equations to get the value of the other variable.
y = 2 × 2 + 1
⇒y = 5
So, (2,5) is the required solution of the pair of linear equations y – 2x = 1 and x + 2y = 12.
Elimination method of finding solution of a pair of Linear Equations
Elimination method
Consider x + 2y = 8 and 2x – 3y = 2
Step 1: Make the coefficients of any variable the same by multiplying the equations with constants. Multiplying the first equation by 2, we get,
2x + 4y = 16
Step 2: Add or subtract the equations to eliminate one variable, giving a single variable equation.
Subtract second equation from the previous equation
2x + 4y = 16
2x – 3y = 2
– + –
———————–
0(x) + 7y =14
Step 3: Solve for one variable and substitute this in any equation to get the other variable.
y = 2,
x = 8 – 2 y
⇒ x = 8 – 4
⇒ x = 4
(4, 2) is the solution.
Cross-multiplication Method of finding solution of a pair of Linear Equations
For the pair of linear equations
a1x + b1y + c1=0
a2x + b2y + c2=0,
x and y can be calculated as
x = (b1c2−b2c1)/(a1b2−a2b1)
y = (c1a2−c2a1)/(a1b2−a2b1)
Solving Linear Equations
Equations reducible to a pair of Linear Equations in 2 variables
Some equations may be in a form which can be reduced to a linear equation through substitution.
2/x+3/y=4
5/x−4/y=9
In this case, we may make the substitution
1/x = u and 1/y = v
The pair of equations reduces to
2u + 3v = 4
5u – 4v = 9
The above pair of equations may be solved. After solving, back substitute to get the values of x and y.
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