Simultaneous Linear Equations  Solving systems of equations graphically
Independent equations, inconsistent equations and dependent equations
System of linear equations word problems
Example:  A two-digit number enlarges by nine when its digits reverse. The same two-digit number divided by sum of its digits gives quotient 5 and reminder 4. Find the two-digit number.
Solution:  Let x be ten's digit and y units' digit, then 10x+ y is a two-digit number.
Then, written are conditions of the given problem: Example:  An isosceles triangle with the sides 6 cm longer then base has the perimeter 48 cm.
What is the length of its base and the sides?
 Solution: Example:  The perimeter of a rectangle is 42 cm. The ratio between its width and the length is 3 : 4.
Find the length and width of the rectangle.
 Solution: Solving systems of equations graphically
The system of two equations in two unknowns can be represented graphically in the Cartesian plane as two
lines l1 and l2.

 l1 ::   a1x+ b1y + c1 = 0 l2 ::   a2x + b2y + c2 = 0 The coordinates (x, y) of the intersection of the two lines are the values of the variables that make both equations true.
Independent equations, inconsistent equations and dependent equations
There are three possibilities:
 a) Independent equations, the lines intersect in one point. There is a unique solution if b)  Inconsistent equations, the lines are parallel but distinct (have the same slope), that is and there is no solution.
 c)  Dependent equations, equations describe the same line, that is thus, there are an infinite number of solutions to the system.
Example:  Solve graphically given system of linear equations:
Solution:

 l1 ::   2x + 3y - 4 = 0 l2 ::   -x + 2y - 5 = 0 Coefficients satisfy the condition: so, the lines intersect. Solve equation y = 0 to get the x-intercept, and x = 0 to get the y-intercept.
Thus, obtained are the points, (2, 0) and (0, 4/3) of the line l1, and (-5, 0) and (0, 5/2) of the line l2.
The lines intersect at (-1, 2).   Intermediate algebra contents 