Download 3x y = 6

Solving Linear Systems
Algebraically
Quiz
1. Solve by graphing:
3x + y = -6
y = 2x – 1
3x + y = –6
y = 2x – 1
You must solve the first
equation for y before you can
graph it
3x + y = -6
-3x
-3x
y = 6 – 3x
Solution
(–1, –3)
Solve systems, substitution
Systems of equations can be solved algebraically rather
than graphically. One method is called substitution.
Example:
2x + 3y = 8 and x + 4y = 9
We are going to solve one of the equations for either x or
y – we must decide which will be easier:
•The easiest thing to solve for is a variable without a coefficient (such as x
in the 2nd equation
•If there is not a variable without a coefficient look for a coefficient that
will divide into the other numbers in the equation without leaving a
fraction
Take the piece in
We will solve the 2nd equation for x:
the circle and
2x + 3y = 8
x + 4y = 9
- 4y -4y
x = 9 – 4y
Now that we have
2(9 - 4y) + 3y = 8
solved it for x we will We will now
solve this
use that in the first
equation for y
equation to solve for y
substitute it in for x
in the equation
putting it in
parenthesis
Example (cont):
So far we have the original 2 equations, the equation we solved for x, and the piece
that we substituted in:
2x + 3y = 8 x + 4y = 9 x = 9 – 4y
2(9 - 4y) + 3y = 8 We now need to solve this equation for y
18 – 8y + 3y = 8 Distribute the 2
We will plug 2 in for
y in either of the
18 – 5y = 8 Combine -8y and 3y
original equations:
- 18
-18 Subtract 18 from both sides
x + 4y = 9
-5y = -10
Divide both sides by -5
x + 4(2) = 9
-5
-5
x+8=9
y = 2 We will use this value to find x
- 8 -8
So our solution to this problem is:
x=1
Solution
(1, 2)
Remember that the solution to a
system is the point that the cross,
so (1,2) would be the point these
two lines cross at if they were
graphed
Example: Solve 2x + 4y = 14 and 2y – 3x = -5
There is no variable without a coefficient so we look at the coefficients and try to
pick one out that will divide into the other coefficients.
In the 1st equation 2 will divide into 4 and 14
without leaving any fractions so we solve for x.
2x + 4y = 14
- 4y -4y
2x = 14 – 4y
2 2 2
x = 7 – 2y
2y – 3x = -5
2y – 3(7 – 2y) = -5
2y – 21 + 6y = -5
8y – 21 = -5
+ 21 +21
Now we plug this in for
8y = 16
x in the 2nd equation
8
8
y=2
We now plug this
into one of the
equations
2x + 4y = 14
2x + 4(2) = 14
2x + 8 = 14
-8 -8
2x = 6
2 2
x=3
Solution – (3,2)
How do we check a solution that we are given or that we find?
Simply plug the alleged solution into both equations – if it
works in both equations then it is a solution…if it doesn’t
work in either or both, then it isn’t a solution.
Is (-2,3) a solution for
the following system: Plug (-2,3) into each
equation and see if it
2x – 3y = -13
works
-3x + 4y = 18
2x – 3y = -13
2(-2) – 3(3) = -13
-4 – 9 = -13
-13 = -13 (true)
-3x + 4y = 18
-3(-2) + 4(3) = 18
6 +12 = 18
18 = 18 (true)
Since (2,3) works in both equations it is
the solution to the system
Find the solution to the following system using the
substitution method and check your answer:
2x – y = 5
4x + 2y = 22
Homework for tonight is 7.2
Remember that the Chapter 5 Extra Credit is due
next Tuesday (the 9th)