Linear Equations & Graphs
Linear equation in one variable
Equation: An equation is a statement of equality of two algebraic polynomial involving one or more variables.
Linear equations: The expression of the form Ax+B, where A and B are real numbers and A ≠ 0, is a linear polynomial and equation involving only linear polynomial are called as linear equations.
- Graph of linear equation of one variable is a straight line, which is either parallel to the horizontal and vertical axis.
- Linear equation in one variables has unique solution.
linear equations examples:
- y = 3x + 1
- 4x = 6 + 3y
- y/2 = 3 − x
linear equations in one variable example:
- 9x + 1 = 5 − x
- t + 3t = 9 − t
Rules for solving a linear equation
- If some number is added to both the sides of an equation, the equality remains the same.
- If same number is subtracted to both the sides of an equation, the equality remains the same.
- If same number is multiplied to both the sides of the equation, the equality remains the same.
- If both sides are divided by a some non-zero number the equality remains the same.
Example: Solve 2(x – 3)-(5 – 3x)=3(x + 1)-4(2 + x) and the value of x is ?
2(x – 3)-(5 – 3x)=3(x + 1)-4(2 + x)
= 2x – 6 – 5 + 3x = 3x + 3 – 8 – 4x
= 5x – 11 = -x -5
= 6x = 6
= x = 1 ( Ans)
Linear equation in two variable :
An equation of the form ax + by + c = 0, where a,b,c are real, a ≠ 0, b ≠ 0 and x and y are variables. It is called as linear equation in two variables.
Example: 2x + 3y = 5, 3√x − y = 6 are linear equation in two variables.
- The linear equation ax + by + c = 0 has an infinite number of solutions.
- The graph of equation ax + by + c = 0 is a straight line so it is called as linear equations.
- Every point on graph of ax + by + c = 0 gives its solution.
Systems of linear equation in two variables.
When we draw the graph of each of the two equations, we have the following three possibilities.
- Either the lines intersect ( Consistent )
- The lines are parallel ( Inconsistent )
- The lines coincide ( Dependent )
Case 1 : When two lines intersect – As two intersecting straight lines always intersect in a point. So there is only one point which lies on both the lines.
If the straight lines in the graph intersect each other at a single point, then system has unique solution.
Case 2 : When two lines are parallel – When two lines are parallel they do not meet at any point so there is no common point on both of the straight lines, So thus system has no solution.
Case 3 : When the lines coincide- When two lines coincide, the points lying on one also belong to the other. Thus all points are common so the system has infinite number of solution.
Consistency of the system of linear equations:
A set of linear equations is said to be consistent, if there exists at least one solution for these equation. A set of linear equations is said to be inconsistent, if there is no solution for these equation.
Let us consider a system of two linear equation as shown.
The system of equations
a1x + b1y + c1 = 0 and
a2x + b2y + c2 = 0
Case 1 no solution if ( Inconsistent )
a1/a2 = b1/b2 ≠ c1/c2
Case 2 an infinite number of solution, if (Dependent )
a1/a2 = b1/b2 = c1/c2
Case 3 a unique solution, if ( Consistent )
a1/a2 ≠ b1/b2
- A solution can be obtained by any of the above four stated method.
linear equation problems:
Question : Which option is correct, for the following pair of equations ?
x + 2y – 4 = 0 and 3x + 6y -12 = 0
(b) Consistent (Dependent)
(d) None of these
Given pair of linear equations is x + 2y – 4 = 0 and 3x + 6y -12 = 0
On comparing with standard form of pair of linear equations, we get
a1 = 1, b1 = 2, c1 = -4
and a2 = 3, b2 = 6, c2 = -12
Now, a1/a2 = 1/3, b1/b2 = 2/6 = 1/3, c1/c2 = -4/-12 = 1/3
a1/a2 = b1/b2 = c1/c2
Hence , the given pair of linear equations is consistent ( dependent ), So option b is correct.
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