Linear Equations

Lesson

Linear Equations may be written in serveral forms:

• Standard form
• Ax+By=C
  
  where, A and B are not both equal to zero and A, B, and C are integers whose greatest common factor is 1.
  
  
• Slope-intercept form
  • y = mx + b
    
    where m is the slope of the line and b is the y-intercept, where the line crosses the y axis.
    

• Point-slope form
  • y - y1 = m (x - x1)
    
    where m is the slope of the line and (x1,y1) is any point on the line. The point-slope and slope-intercept forms are easily convertable.

   

Review of Linear Equations:

Slope = m = rise/run = change in y / change in x = (y2 - y1) / (x2 - x1)

A positive slope moves up and to the right:

A negative slope moves down and to the right:

In Finding the Intersection Point of Two Lines, you Solve the Equations for X and Y.

You may solve by Substitution or Elimination:

Substitution:

2x + 3y = 4

4x + oy = 8

The steps of substitution:

OOOOOSolve one of the equations for one of the variables, x or y, and then substitute that value for the variable.

2x + 3y = 4
4x + y = 8 becomes y = -4x + 8
___ __SOooo
____2x + 3(-4x + 8) = 4
iiiiiiiiiiiiiiii Then by distributing,
ooooooooooooo 2x + -12x + 24 = 4
Then solve for x by getting all the variables on
oo one side and the numbers on the other.
00000000000000000000000 -10x = -20
ooooooooooooooooooooTHEN divide by -10
00000000000000000000000000 x=2
0000000000 THEN plug the x back into
ooooooooooooooooooo oooooy = -4x + 8
ooooooooooo ioooooooooooooy = -4(2) + 8
0000000000000000000000000 Solve for y,
oooooooooooooooooooooooooooo y =0

So the two equations intersect at the point (2, 0) OOOOO

Elimination:

2x + 3y = 4

4x + y = 8

To do elimination, get the coefficients of one of the variables, the numbers in front of the x and y, to be the same, and then subtract the equations.

2x + 3y = 4
_Multiply the above equation by 2 so the x variable will cancel
4x + y = 8
oooooooo0000 2 ( 2x + 3y = 4)
and then subtract the equations
4x +6y = 8
- (4x + y = 8)
5y = 0
y = 0
Plug 0 in for y and 2x + 3 (0) = 4
2x = 4
x=2
Again the two equations intersect at the point (2, 0)OOOOOOOO

Upon solving the problem, one of these things will happen:

• If you get one point (x,y) for your answer, then there is one solution.
• If you find that the lines are the same, there are an infinite number of solutions.
• If you find no values work for x and y, then there is no solution.