Systems of Equations

1
Systems of Equations
A system of equations occurs when 2 or more
relations containing 2 variables are presented
together for comparison.
These relations can be presented in a variety of
forms.
By algebraic description x 2y 6 y 3x - 4
By graph
By listing their elements in a set S1
(-4,1), (-2,1), (0,5), (1,7), (2,9) S2
(-3,1), (-2,1), (-1,1), (0,1), (1,1)
The solution to a system of equations is
basically the ordered pair(s) that are shared by
both equations.
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To determine the solution of a system presented
on a graph, it is necessary to identify the
points of intersection.
f ? g (2,-7)
f ? h (-7,2), (1,-6)
g ? h (-1,-7)
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Systems of Equations of the 1st Degree
When the exponent attached to the variables of
the equations is 1, we say that the equations are
first degree equations. When these are
represented on a graph they form a straight line
and we therefore call them linear equations.
There are 2 conventional formats that linear
equations may be presented.
Slope-intercept form y mx b
Standard form Ax By C 0
y x
2x - y 5 0
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The solution to a system of equations can be
determined algebraicly or graphically. The
following table identifies 3 different scenarios
that can arise with systems of linear equations.
Unique S (x,y)
No Solution S Ø
Infinite Same as equation
Nature of Solution
Characteristics of the Equations
A1x B1y C1 A2x B2y C2
y m1x b1 y m2x b2
m1 m2 b1 ? b2
m1 m2 b1 b2
m1 ? m2
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The following table identifies different
scenarios that can arise with systems involving a
linear equation with a quadratic equation.
Secant System Intersecting at 2 points
Non-intersecting or Disjoint
Intersecting at 1 point
No Solution S
1 Solution S (1,-2)
2 solutions S (-3,-2),(1,2)
Nature of Solution
Scenarios
Oblique or horizontal line tangent to parabola
Oblique or horizontal line secant to parabola
Linear graph below parabola opening up
Vertical line intersecting parabola
Linear graph above parabola opening down
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Horizontal line secant to parabola
Oblique line secant to parabola
Linear graph below parabola opening up
Linear graph above parabola opening down
Horizontal line tangent to parabola at its vertex
Vertical line intersecting parabola
Oblique line tangent to parabola
7
Solve the following system of equations by
constructing a graph. 2x y 9 0 y x2
4x 9
One of these equations is a linear equation. To
graph a linear equation, we can isolate y and
make a table of values to find ordered
pairs. 2x y 9 0 -y -2x
9 y 2x 9
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The other equation is a quadratic equation. To
graph a quadratic equation, we must find the
vertex and other points.
y x2 4x 9 a 1 b 4 c 9 ? b2
4ac 42 4(1)(9) 16 36 -20
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There are 2 solutions for this system oblique
line secant to a parabola. In this case, the
solutions can be found by just observing both
tables of values.
However, this will not often be the case. You
can determine the solutions by identifying the
ordered pairs for which the straight line and
parabola intersect (-2,5) and (0,9).
S (0,9), (-2,5)
This is a secant system because the line
intersects the parabola at two points.
10
Solve the following system of equations by
constructing a graph. x 2y - 6 0 y -x2
4x - 3
11
Solve the following system of equations by
constructing a graph. x 2y - 6 0 y -x2
4x - 3
y -x2 4x - 3 a -1 b 4 c -3 ? b2
4ac 42 4(-1)(-3) 16 12 4
S
This is a disjoint system because there is no
intersection.
12
Solve the following system of equations by
constructing a graph. x - y 6 0
y x 6
13
S (-5,1)
This is a tangent system because the line
intersects the parabola at one point.