#
Section 5-2

#
Solving Linear Systems by Substitution

A
system of linear equations
**
**
is just a set of two or more linear equations.

In two variables, the graph of a system of two equations is a pair of lines in the plane. These lines will intersect in either zero points (if they are parallel), one point (in most cases), or infinitely many points (if the graph of the two equations is the same line).

To solve a linear system by substitution:

1) first solve one equation for one variable - say, get
*
y
*
alone on one side, with some expression involving
*
x
*
on the right side.

2) Then, take the second equation - the one you haven't worked with yet - and rewrite it, using the expression you got in the last step instead of
*
y
*
.

3) Solve the new equation for
*
x
*
. This is the
*
x
*
-coordinate of the point where the two lines meet.

4) You can then substitute in either equation to find
*
y
*
.

Example:

Solve by susbtitution.

*
y
*
=
*
x
*
**
–
**
4

**
3
***
x
*
+ 2
*
y
*
= 7

We're lucky here: the first equation already has
*
y
*
alone on the left side.

So, just substitute
*
x
*
**
–
**
4 for
*
y
*
in the second equation:

**
3
***
x
*
+ 2(
*
x
*
– 4) = 7

Now simplify and solve for
*
x
*
.

**
3
***
x
*
+ 2
*
x
*
– 8 = 7

**
5
***
x
*
= 15

*
***
x
**
**
= 3
**

Finally, substitute
*
x
*
= 3 in either of the original equations (we'll use the first one because it's easier).

*
y
*
= 3
**
–
**
4

**
***
y
*
= –1

So, the solution is
**
(3, –1)
**
. This is the point where the two lines intersect.

If something breaks down along the way:

It's possible for this method to fail; you may get redundant equations, or equations that don't make sense. If this happens, you have a pair of lines which are parallel, or two equations which represent the same line. Check your slopes and

*
y
*
-intercepts
!