Systems of Equations
1
System of Linear Equations
A system of linear equations is a mathematical way of saying more than one line. In a system of linear
equation we are usually trying to determine what points the equations have in common. A system of linear
equations can have three different interactions – One solution, No solution and Infinite solutions.
One Solution (Intersection)
No Solution (No Intersection)
Infinite Solutions (Same Line)
Different Slopes
Equal Slopes, Different y intercepts
Equal Slopes, Same y intercept
One Solution
y = 2x – 3
y = 5x + 8
2
y  x 4
3
y  3x  2
No Solution (No Intersection)
Infinite Solutions (Same Line)
y = 3x + 8
2y = 6x + 8
y = 3x – 1
y = 3x + 4
5 y   x  30
5
y   x 4
1
4
y x  6
5
5
y   x 1
4
There are typically three different ways that are used to solve for the solution(s) of a system of equation and
each method has its own strengths and weakness.
Technique 1 – Graphing
Strengths of this Technique:
Weakness:
Graphing is a great technique to use to get a general idea of what is going on
and what types of values you can expect but is not a very accurate method.
When we graph we expect values to come out as integers and so it is difficult to
determine values that are not integers. For example in the graph on the right,
what values would you put for this system of equation?
(3,5)?
(2.5, 4.5)?
(2.7,4.8)?
Really this method is quite limited but it is a very powerful tool to help you
understand the expected result.
Systems of Equations
2
Technique 2 – Substitution
Strengths of this Technique:
Weakness:
Because we are looking for the single point where both lines equal each other, the x and y values are the same
value at that point for both lines. Each line has an infinite amount of points but together they only have one
common point when they intersect. This allows us to use the x value of one line as the same x value in the
other line and the same for y. One way to solve two equations with two unknowns is to substitute a variable
value into the other equation. This is a method that we use when we can easily isolate the x or y variable.
Examples: Determine the intersection of the two lines
a)
(1) y = 3x + 8
(2) 2x + y = -2
Equation (1) already has an isolated variable and
so we can substitute its value into equation (2).
b)
(1) 5y = x + 1
(2) 2x + 3y = 11
In equation (1) we can isolate variable quite easily.
(1) x = 5y – 1 and so we can substitute its value into
equation (2).
So (____, ____) is the intersection of these two
lines.
So (___,____) is the intersection of these two lines.
Systems of Equations
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Technique 3 – Elimination
Strengths of this Technique:
This technique works nicely when the variables don’t isolate very easily. The side benefit of learning this
technique is that it prepares students for matrices and matrix operations. The idea of this method is to alter
the equation or equations through multiplication so that one of the variables cancels either through addition
or subtraction.
Examples: Determine the intersection of the two lines
a) (1) 2x – 3y = 3 (2) 4x + 3y = 15
b) (1) 5x – 2y = 4 (2) 2x + 4y = 17
c) (1) 2x – 6y = 8 (2) 3x + 5y = -2
In this case, we did not have to
alter the equations because the y
values canceled naturally.
This time we needed to multiply
equation (1) by two so that they y
values would cancel.
This time we needed to multiply
both equations by different values
to create a variable that would
cancel. We multiplied equation (1)
by three and equation (2) by
negative two.
Determine if the following systems of linear equations have (O)ne Solution, (N)o Solutiion or (I)nfinite
Solutions.
a)
b)
c)
1
y x3
2y  x  4
y   2x  5
3
1
y  5x  2
1
y x4
y x 5
2
3
O or N or I
O or N or I
O or N or I
Systems of Equations
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Determine which of the technique would be the simplest to use based on the initial equation forms,
Substitution or Elimination. Why did you pick it?
a)
b)
c)
d)
x  1y  5
y  8x  5
17  2 x  5 y
y  3
22  2 x  y
y  7x  4
7   2x  3y
y   5x  3
Substitution or Elimination
Why?
Substitution or Elimination
Why?
Substitution or Elimination
Why?
Substitution or Elimination
Why?
System of General Equations
A system of equations does not always have to be with lines. A system of equations could be a line and a
circle, a line and a parabola, two circles, etc… the list can go on and on. Ultimately, the process is the same as
what we did with two lines.
A Circle and a Line
Two Circles
A Parabola and a Line
Let us look at these examples up close.
a) Determine the intersections of the circle
2
2
 x  2    y  2   16 and the line x = 5.
b) Determine the intersections of the circle  x  5   y 2  49
I am going to use substitution because x = 5
is an already isolated variable.
I think I will use substitution here as well after I alter the first
2
equation, y 2  49   x  5 
2
and the circle  x  4   y 2  16 .
2
Systems of Equations
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a) Determine the coordinate(s) of any points of intersection
2
between the circle ( x  1) 2   y  2   9 and
the vertical line, x = 4.
c) After determining the points of
intersection, now graph the circle and
the line to confirm your calculations.
b) Is x = 4 a tangent or a secant to the circle? _____________
HOMEWORK
1. Determine if the following systems of linear equations have (O)ne Solution, (N)o Solutiion or (I)nfinite
Solutions.
a)
b)
c)
8 y  6 x  16
3
y x2
4
3 x  5 y   10
5
y x2
3
2x  2 y  2  0
y  1x  1
O or N or I
d)
O or N or I
e)
f)
3
x 1
7
7
y x4
3
2x  y  1
4 x  2 y  8
y
x6
y  3x  5
O or N or I
O or N or I
O or N or I
O or N or I
Systems of Equations
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2. Determine which of the technique would be the simplest to use based on the initial equation forms,
Substitution or Elimination. Why did you pick it?
a)
b)
c)
d)
10  4 x  6 y
7  2 y  5 x
11  8 x  5 y
y  2 x  12
9   3 x  3 y
2  5x  7 y
18  7 x  4 y
24  4 x  y
Substitution or Elimination
Why?
Substitution or Elimination
Why?
Substitution or Elimination
Why?
Substitution or Elimination
Why?
3. Solve the system of equations. (Use the technique you chose above)
a)
10  4 x  6 y
9   3 x  3 y
b)
7  2 y  5 x
2  5x  7 y
c)
11  8 x  5 y
18  7 x  4 y
d)
y  2 x  12
24  4 x  y
4. Graphing is one of the techniques that we can use to determine the solution for a system of equations.
Solve the given system using graphing and then use one of the other techniques to solve it again to check
your graphing answer. Explain why graphing is not a very accurate technique.
a) Graph
b) Solve
3
3
y  x 1
y  x 1
4
4
y  2 x  7
y  2 x  7
Solution:
( ______ , ______)
c) Why is graphing not a very accurate
technique?
Systems of Equations
5. a) Determine the coordinate(s) of any points of intersection
2
between the circle, ( x  2)2   y  5   25 and the line, x = 1.
7
c) After determining the points of
intersection, now graph the circle and
the line to confirm your calculations.
b) Is x = 1 a tangent or a secant to the circle? _____________
6. a) Write the circle, x2 + y2 + 4x -2y -20 = 0, in center/radius form.
Equation in center/radius form
__________________________
Center ( ______ , ______ )
Radius = __________
b) Determine the coordinate(s) of any point of intersection
between the circle and the line, y = 3x + 2.
c) After determining the points of
intersection, now graph the circle
and the line to confirm your
calculations.