System of Equations

2 (or more) equations,
each of which has 2 (or
more) variables
Our goal is normally to find a
the values of the variables that
work in both equations.
In most cases you can think of
this as the point where two
lines cross.
(x,y) is the
solution to
the system
of equations.
There are lots of ways to
solve systems of equations.
One way is by graphing.
 Carefully graph both lines
on the same axes.
 Find the point where they
cross.
Solve y = 2x + 2
y=x–1
Solve y = 2x + 2
y=x–1
y = 2x + 2
 y-intercept = 2
2
 slope = /1
y=x–1
 y-intercept = -1
 slope = 1/1
The
solution
is
(3, -4)
Solve y = x + 2
y = -x + 4
y=x+2
 y-intercept = 2
1
 slope = /1
y = -x + 4
 y-intercept = 4
 slope = -1/1
The
solution
is
(1,3)
Solve x + 2y = 5
2x + y = 4
Solve x + 2y = 5
2x + y = 4
Find the intercepts.
Solve x + 2y = 5
2x + y = 4
Find the intercepts.
x + 2y = 5
(0,2.5) and (5,0)
2x + y = 4
(0,4) and (2,0)
The
solution
is
(1,2)
Solve 2x + 2y = 6
4x – 6y = 12
Solve 2x + 2y = 6
4x – 6y = 12
Intercepts
2x + 2y = 6
(0,3) and (3,0)
4x – 6y = 12
(0,-2) and (3,0)
We can tell without even
finishing the graph that (3,0) is
the solution.
Solve 3x + 2y = 12
3x + 2y = 6
Solve 3x + 2y = 12
3x + 2y = 6
Intercepts
3x + 2y = 12
(0,6) and (4,0)
3x + 2y = 6
(0,3) and (2,0)
These are
parallel lines.
They both have
a slope or -3/2.
They never intersect, so there
is no solution.
It’s also possible to have
infinitely many solutions,
which would mean you have
different expressions for the
same line.
For example: 5x + 2y = 10
-5
y = /2x + 5
There are lots of word
problems that involve systems
of equations. For instance …
Two tacos and a burrito cost
$8. One taco and two
burritos cost $10.
How much is a taco,
and how much is a
burrito?
Two tacos and a burrito cost
$8. One taco and two burritos
cost $10. How much is a taco,
and how much is a burrito?
2t + 1b = 8
1t + 2b = 10
2x + 1y = 8
1x + 2y = 10
2x + 1y = 8
1x + 2y = 10
The solution is (2,4)
Two tacos and a burrito cost
$8. One taco and two burritos
cost $10. How much is a taco,
and how much is a burrito?
(2,4) means a taco costs $2
and a burrito costs $4.