top of page
EXAMPLES & SOLUTIONS
System of Equations:  Graphing on the Same Line
Anchor 1

They form two linear lines that coincide throughout the entire system of equations.

Same lines have infinite solutions.
The points intersect at every point along both linear equations.
They form solutions for every point.
The slope and y-intercept are exactly the same.
GRAPHING FROM THE SAME LINE
Explanation of how a system of equations form on the same line
Slope-intercept
     y  =  mx  +  b
y  =  x  +  2
Standard Form
     Ax  +  By  =  C
  -3x  +  3y  =  6
FIRST STEP: CREATE INITIAL LINEAR LINES
SLOPE-INTERCEPT TECHNIQUE
Discover the y-intercept from the slope-intercept equation
Equation #1
Slope-Intercept
y  =  mx  +  b
y  =  x + 2
Discover the y-intercept
          y =  x + 2     
      if x = 0, then
          y = (0) + 2
          y = 0 + 2
           y = 2

     y-intercept 
              (0, 2)
Equation #2
Standard Form
Ax +  By = C
-3x + 3y = 6
Standard Form - Slope-intercept
            -3x + 3y = 6 
-3x  + 3x  + 3y = 3x + 6
                        3y = 3x + 6                                              3y = 3x + 6   
          
                      y = x + 2

                 

                 if x = 0, then
                       y = 0 + 2
                       y = 2

                y-intercept 
                      (0, 2)
3       3      3
Discover the y-intercept
SECOND STEP: DISCOVER THE SLOPE
SLOPE-INTERCEPT TECHNIQUE
Discover the slope from the slope-intercept equation
Discover the slope
                   y = x + 2
                y = mx + b

          y-intercept = b,                                    therefore, 
 y = b = 2 =  y-intercept (0, 2)       
                    then,
          coefficient of x = 1,
              slope =  m =  1          
Discover the slope
          -3x + 3y = 6 
         transform to:  y = m + b
-3x  + 3x+ 3y = 3x + 6
                       3y = 3x + 6                                            3y = 3x + 6   
           
                       y = x + 2
                    y = mx + b

              y-intercept = b,                                    therefore,   
     y = b = 2 =  y-intercept (2)     

                       then,       
              coefficient of x = 1,                             slope =  m =  1   
3       3      3
Draw the slope for each graph. This assist in ensuring the right linear direction.
Equation #1: 

4

-4

-2

1 unit

4

-2

 (0, b) = (0, 2)

1 unit

2

2

P (1, 3)

Equation #2: 

4

-4

-2

1 unit

4

-2

 (0, b) = (0, 2)

1 unit

2

2

P (1, 3)

SOLUTION
Slope and y-intercept are a good way to quickly observe system of equations.

Rule:
Both slopes are equal
Both y-intercepts are equal

Place linear lines

-2

4

-4

4

-2

 (0, 2)

2

2

EQ. 1

EQ. 2

m = 1

Equation #1

y  =  x + 2

m = 1

Equation #2

-3x + 3y = 6

Equation #1
slope =  1
y-intercept = (0, 2)
Equation #2
slope =  1
y-intercept = (0, 2)
Solutions: Infinite

Equations that are on the same line have points that connect throughtout both linear lines.

The slope and y-intercepts' are equal.
The two linear lines form a dependent system of equations.
The y-values depends on the x-values throughout the both linear lines.
This means that there are an infinite number of solutions.
Therefore, the system of equations are consistent.
Graphically these linear lines  INTERSECT
at
every point
therefore, there are
infinite solutions

You are observing
same linear lines.
bottom of page