General Contents
Detailed Contents
Index
Programmed tutorial: Elimination Method for Linear Systems: Example 2
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Use the
Elimination Method
to solve
5
x
4
y
= 2
Equation 1
2
x
3
y
= 5
Equation 2
[Here is a
substitution method version
.]
General Contents
Detailed Contents
Index
Should we try to eliminate
x
or
y
?
It doesn't matter!
Let's eliminate
y
this time. What multipliers do we need to use?
Since we want to end up with the same coefficient on y in each equation, we can use a multiplier of 3 in the first equation and a multiplier of 4 in the second equation.
Why is one multiplier negative?
We want to eventually add the modified equations and have
y
drop out. With these multipliers, one equation will have a term 12
y
and the other will have a term +12
y
.
When we multiply, does only the coefficient of
y
change?
Recall our definition of equivalent equations. To preserve an equation, we have to multiply
all terms on both sides
by the same factor.
Obtain an equation equivalent to the first equation by multiplying by 3.
3*(5
x
4
y
) = 3*(2).
Do the multiplication.
We get
15
x
12
y
= 6
Multiply the second equation by 4.
( 4)*(2
x
3
y
) = (4)*( 5)
Do the multiplication.
8
x
+ 12
y
= 20
Add these two modified equations:
15
x
12
y
= 6
8
x
+ 12
y
= 20
Adding, we get
7
x
= 14.
Divide by 7 on both sides.
x
= 14/7 = 2
Are we done?
No. We are solving for an (
x
,
y
) pair.
What do we do next?
We substitute this value for x in one of the original equations.
Do that in equation 1.
Substituting
x
= 2 into 5
x
4
y
= 2, we get
5(2) 4
y
= 2, or
10 4
y
= 2.
Subtract 10 from both sides.
4
y
= 2 10 = 12.
Divide by 4 on both sides.
Now state the (
x
,
y
) pair which is the solution for this system of simultaneous equations.
(
x
,
y
) = (2, 3)
We now check
x
= 2,
y
= 3 in
both
of the original equations.
In Equation 1:
5(2) 4(3) = 10 12 = 2 OK
In Equation 2:
2(2) 3(3) = 4 9 = 5 OK.
Summarize these results.
The point (2, 3) solves the original equations.
If we plot the two lines represented by the original equations, which line will (2, 3) be on?
It will be on both lines, at their intersection.
Plot these lines to see this. Check your graph by clicking Next.
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