SAT Algebra: Understanding Linear Equations in Two Variables
Here we attempt to understand ‘Linear Equations in Two Variables’. To recapitulate, let us first understand what we mean by equating with one variable –
- In mathematics, a variable is a value that may change within the scope of a given problem or set of operations.
- In contrast, a constant is a value that remains unchanged, though often unknown or undetermined.
- A real number may be either rational (44 or −29/129) or irrational numbers (pi and square root of 2); either algebraic or transcendental; and either positive, negative or zero. In simpler words, Real numbers can be thought of as points on an infinitely long number line.
Linear Equation in One Variable
If a, b, c are real numbers and x is a variable, then an equation of the type ax + b=c (where a in not equal to 0) is called a linear equation in one variable. On substituting a real number for the variable (x) in a given equation (ax+b), if both the sides of the equation become equal, then the number is called the solution of the equation.
Hence, to solve an equation means to find its solution.
- Two or more equations are said to be equivalent equations when a solution of one equation is also the solution of the other.
- A number which is not a solution of the first is also not the solution of the other.
E.g.: x+3=5 and x+9= 11 are equivalent equations.
Equivalent equations can be obtained by performing the following four basic operations on a linear equation.
a) By adding the same number on both sides.
b) By subtracting the same number on both sides.
c) By multiplying both sides by the same non-zero number.
d) By dividing both sides by the same non-zero number.
After performing the above operations, the final form of the equivalent equation is x = k (k being a real number). The solution of the first equation will also be k, because it is the solution of the later.
A Linear Equation in Two Variables
If a, b, c are real numbers (a and b not equal to 0), then an equation of the type ax + by=c is called a linear equation in two variables.
Here variables are x and y.
Example: Each of the following is a linear equation in two variables:
1) 2x – y = 7
2) x – 3y = 8
3) x/4 – y/3 = 5
4) 3m + 2n = 2
On substituting a pair of real numbers for the two variables(x and y) in the given equation, if both sides of the equation become equal, then the pair of real numbers is called the solution of the equation.
X= 4 and y=1 is the solution of the equation 2x – y = 7, because on substituting x =4 and y = 1 in the equation, we have,
2*4-1 = 7
Hence, 7= 7 this is true.
Similarly, x=5 and y= -1 is the solution of the equation x-3y =8.
SAT TIP 1:A simple linear equation is an equality between two algebraic expressions involving an unknown value called the variable
SAT TIP 2: A linear equation in two variables has infinite number of solutions. In other words, it does not have one unique solution.
x = 1, y = -1 is the solution of 2x + 3y = 5
2(1) + 3(-1) = 5 and
x = -2, y = 3 is also a solution of 2x + 3y = 5
8(-2) + 3(3) = 5
-4 + 9 = 5
Similarly, we can find many solutions for 2x + 3y = 5
SAT TIP 3: In an equation of two variables ax +by = c, if
a = 0 (but, b is not equal to 0) or
b = 0 (but a is not equal to 0)
then it reduces to the form by=c or ax = c
I.e. y= c/b
In other words, y=k1 or x=K2
An excellent characteristic of equations in two variables is their adaptability to graphical analysis. The rectangular coordinate system is used in analyzing equations graphically.
- The graph of a linear equation in two variables is always a straight line which is neither a horizontal line nor a vertical line ( except when in an equation a=0 or b=0)
- Every point on this line represents a solution of the equation
- If a pair of values when substituted for the two variables of an equation does not give the solution of the equation, the point formed of these values does not lie on the graph of the equation.
- Graph of an equation can be obtained even if only two points are plotted on the graph paper.
Algebraic methods of Solving Linear Equations
The most common algebraic methods of solving 2 equations in two variables are
a) Elimination Method
b) Substitution Method
Elimination Method: In the elimination method, we eliminate one of the two variables x and y either by addition or subtraction, provided that the coefficients of one of the variables in each equation are equal.
If the coefficients are equal and opposite in sign, we eliminate by addition
If the coefficients are equal and have the same sign, we eliminate by subtraction.
Example: Solve x and y
X+ 5y = 34……… (1)
x- 5y= -6…….. (2)
By Elimination: Since the coefficients of y are equal and opposite in sign, we can eliminate by adding both sides of (1) and (2).
2x = 28 or x = 14
Putting x = 14 in (1), we get
14+ 5 y = 34 or 5y = 20 or y = 4.
Substitution Method: In this method we express one of the variables in terms of the other variable from either of the two equations. When this expression is put in the other equation, an equation in one variable is obtained.
From this equation in one variable, we determine the value of this variable, and putting it in either of the two question, we get an equation in the other variable. The value if the other variable is obtained from this question.
Using the same example above in substitution:
From (1), x = 34 – 5y
Substituting 34- 5y for x in (2), we get
34 - 5y – 5y = -6 or 34 – 10y = -6
Or 10y = 40 or y= 4.
From (1), x+5*4 = 34
X+20=34 or x=14
Thus, x=14 and y=4 are the required values.