Systems Of Equations (in two variables x,y)
Systems Of Equations: A system of equations is simply more than just 1 equation.
Solution(s) to a system: is ordered pair(s) of numbers that satisfy both equations (is a
solution for both equations).
Example: For the following 2 equations point (2, 3) is a solution for the system.
Equation 1:
y=-x+5
(3) = - (2) + 5
3 = 3 (a solution)
Equation 2: y = x + 1
(3) = (2) + 1
3 = 3 (a solution)
Since point (2, 3) checks out as a proper solution for both equations
it is said to be a SOLUTION TO THE SYSTEM.
Determining System Solutions Graphically:
A) LINEAR SYSTEMS: two or more linear equations, can have------0 Solutions
(called Inconsistent)
Parallel Lines
Equations with the
SAME SLOPE, but
DIFFERENT
Y-INTERCEPTS
Ex. y = -2x + 3
y = -2x + 7
1 Solution
(called Consistent)
Equations with
DIFFERENT SLOPES,
but the y-intercepts could be
the same or different.
Ex. y = 5x + 3
y = -2x + 7
Infinite Solutions
(called Dependent)
Equations with the
SAME SLOPE
and
SAME Y-INTERCEPT
Ex. y = -2x + 3
y = -2x + 3
B) QUADRATIC LINEAR SYSTEMS:
a quadratic and a linear equations, can have------0 Solutions
1 Solution
2 Solutions
C) LINEAR INEQUALITY SYSTEMS:
a linear inequality system, can have------Infinite Solutions
No Solutions
Steps for Graphing a System of Equations
&
Determining Solutions
1) Make sure all of your equations are set equal to “y”
(in other words, they are in proper standard form).
2) Graph each graph on the SAME coordinate grid.
3) Identify the point(s) of intersection….”THESE ARE YOUR SOLUTIONS”…and
identifying them is the entire reason for you graphing in the first place.
4) Check to make sure the solutions are correct by substituting them back into the
equations/inequalities and solving.
Reminders for Graphing:
A) Linear Equations:


y = mx + b
Plot your y-intercept (b) first on the y axis.
Use your slope (m) to plot more points on the graph. Remember slope is a ratio,
where the top number is your vertical move and your bottom number if you
horizontal move.
B) Linear Inequalities:
y  mx + b
Plot your line just as you would for a linear equation, with the following added
pieces.
1) Dashed Line if the symbols are  or .
2) Solid Line if the symbols are  or .
3) Shade “below the line if the symbols are  or .
4) Shade “above” the line if the symbols are  or .
5) If you do not know whether to shade above or below the line using the
symbols, use a “test point” to make your decision before shading.
C) Quadratic Equations:





y = ax2 + bx + c
Determine your axis of symmetry using the formula x=-b/2a.
Substitute the x you found into the quadratic equation to find the y. The (x, y) pair
is the turning point and the crucial “middle” point of your parabola.
Pick 3 values higher than the x you found and 3 values lower. This will help you get
the full parabola shape.
Substitute these values into the quadratic equation to find their y partners.
Graph all 7 seven points on the graph paper.
On all the graphs DO NOT FORGET TO…..
1) Place arrows on the end of your graphs.
2) Label each graph with the proper equation.
3) Clearly label all solutions.
Your TI-84 GRAPHING CALCULATOR can also create the graph
and the x,y table of values you need to graph these equations and
inequalities on the graph paper.
The only thing you will not be able to get from the graphing
calculator is the shading for the inequalities. Although, this is an
APP for for shading, it is NOT ALLOWED on the regents exam.