Transcript elimination method

Solving Linear Systems
by Linear Combinations
Lesson 7.3
Try adding this
linear system together.
4x + 3y = 1
{
2x – 3y = 1
Notice that the “y” terms in both equations
are opposites of each other.
If we add this system together, the “y” terms
will cancel out and we can solve for “x”.
4x + 3y = 1
+ 2x – 3y = 1
6x + 0 = 2
x = 1/3
Now that you solved for “x,” substitute it
back into either equation and solve for “y.”
2(1/3) – 3y = 1
-3y = 1/3
y = -1/ 9
Solution
(1/3, -1/9)
Important Steps:
1. Arrange equations with like terms in
columns.
2. Study the x
or
y coefficients.
Multiply one or both equations by an
appropriate number to obtain new
coefficients for x or y that are
opposites.
3. Add equations. One of the
variables will disappear. Solve
for the remaining variable.
4. Substitute the value in Step 3 into either
of the original equations and solve for the
other variable. You now have an ordered
pair.
5. Check the solution in each
original equation. If both are
true, you have a solution.
3x + 5y = 6
-4x + 2y = 5
What can we multiply by so that either the x or y
value disappears?
The top equation can be multiplied by 4 and the
bottom equation by 3 to end up with 12x and –12x.
When added together the x’s disappear. Solve for y.
12x + 20y = 24 (multiplied by 4)
-12x + 6y = 15 (multiplied by 3)
26y = 39 (combined, now divide by 26)
y = 3/2 (y value, plug in to find x value)
3x + 5(3/2) = 6 (y value plugged in)
3x + 15/2 = 6 (now subtract 15/2)
3x = -3/2 (now multiply by 1/3)
x=-½
Solution: (- 1/2, 3/2)
Check: plug back into the
original equations.
* Tip - When there are fractions in the
equations, clear the fractions first!
When to use which method?????
Graph: to approximate a solution, check a
solution or show visual.
Substitution or combination: for exact solutions.
Substitution: when one variable has 1 or –1 for
the coefficient.
Combination: when coefficients are larger.
Final Practice Problem…
8x + 2y = 16
5x – y = 28
8x + 2y = 16
10x – 2y = 56
18x = 72
x=4
Multiply the 2nd equation by 2 to cancel
out the “y’s.”
8(4) + 2y = 16
2y = -16
y = -8
Solution: (4, -8)