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Solving Systems of Linear
Equations using Substitution
Substitution
Any system of linear equations in two variables can be solved
by the substitution method.
This method lends itself to situations in which one equation is
already solved for either x or y.
Substitution
x = 3y + 2
2x - 4y = 10
Notice that the first equation gives us an alternate way to
express x (3y + 2).
Let’s re-write the second equation with the “new name” for x.
2( 3y + 2 ) – 4y = 10
Solve for the First Variable
2( 3y + 2 ) – 4y = 10 Distribute
6y + 4 – 4y = 10
2y + 4 = 10
2y = 10 – 4
2y = 6
y=3
Solve for the Second Variable
From our first equation, we know:
x = 3y + 2
We just found that y = 3, so we can substitute this information
into the equation to solve for x.
x = 3( 3 ) + 2
x=9+2
x = 11
Answer is an Ordered Pair
We have found:
x = 11 and y = 3
Our answer is the point of intersection of the two lines:
( 11, 3 )
Remember the BIG Picture
When we are solving systems of simultaneous linear equations,
we are actually looking for the point of intersection of two
lines.
Although a logical way to do this is by graphing, sometimes the
numbers do not lend themselves very well to the graphing
technique. It is almost impossible to read the point of
intersection from a graph when fractions are involved.
Substitution will always work.
3 Cases
Just remember that there are always three possibilities when
you are looking for the point of intersection of two lines.
• The lines can intersect in a single point.
• The lines can be parallel and not intersect at all.
• The lines can live one on top of the other with an infinite
number of points of intersection.
Intersecting Lines
If a system has one, or more solutions, it is said to be consistent.
If the equations represent two different lines, the equations are said
to be independent.
If the lines intersect:
the system of equations is consistent.
the equations are independent.
there is exactly one solution – an ordered pair
the solution will be the point of intersection (x, y)
Intersecting Lines
If the lines intersect in a single point, when you use the substitution
method to solve the equations you will get a number for x and a
number for y.
Since these numbers represent the point of intersection of the two
lines, they should be written as an ordered pair.
In our example, the answer should be written (11, 3).
Some authors, however, simply write x = 11 and y = 3.
Coincident Lines
When you are solving a system of equations using the algebraic
method of substitution, if you get an identity (statement that’s
always true), then the lines are coincident (one on top of the other).
y = 2x
5y = 10x
5(2x) = 10 x
10x = 10x
Coincident Lines
If a system has one, or more solutions, it is said to be consistent.
If the equations represent the same line, the equations are said to be
dependent.
If the lines are the same (coincident):
the system of equations is consistent.
the equations are dependent.
there are infinite solutions – all the points on the lines
sometimes the solution will be expressed in set notation.
{(x, y)|y = 2x}
Parallel Lines
When you are solving a system of equations using the algebraic
method of substitution, if you get an contradiction (statement
that’s never true), then the lines are parallel.
y = 2x
6x – 3y = - 3
6x – 3( 2x ) = - 3
6x – 6x = - 3
0=-3
The two equations represent parallel lines.
Parallel Lines
If a system has no solutions it is said to be inconsistent.
If the equations represent two different lines, the equations are said
to be independent.
If the lines are parallel:
the system of equations is inconsistent.
the equations are independent.
There is no solution.
In set notation, it is the empty set.
Systems of Linear Equations
If you substitute the information from one
equation into another, and you can solve for one
variable, then use that value to solve for the other
variable. That’s your point of intersection.
If you get an identity, the lines are coincident.
If you get a contradiction, the lines are parallel.