Download Solving Linear Systems by Linear Combinations

Section 3.3
Goal: Solve a system of linear equations in two variables by the linear
combination method
SOLVING LINEAR SYSTEMS BY LINEAR
COMBINATIONS
WARM-UP
1. Solve the system by substitution: x – 2y = 16
2x + y = 12
(8, -4)
2. The home team fans for a football game bought five times as many tickets as the
visiting team fans. The total number of tickets sold was 1440. How many tickets
did the home team and the visiting team fans buy?
The home team bought 1200 tickets and the visiting
team bought 240 tickets
3. Write the equation of the line that goes through (2, -1) and (1, 3)
y = -4x + 7
USING THE LINEAR COMBINATION METHOD
EXAMPLE 1: SOLVE THE SYSTEM
x y 7
2x  y  2
3x  9
x 3
3 y  7
y4
2(3)  4  2
64  2
The coefficient of the y in Equation 1 is 1.
The coefficient of the y in Equation 2 is -1.
They are opposites so you do not need to
multiply by anything! Add the two equations
together.
Now substitute 3 for x in either of the two
equations.
Therefore the solution is (3, 4).
Check to see if (3, 4) works in the other equation.
It works! So (3, 4) is the point where the two lines intersect.
EXAMPLE 2: SOLVE THE SYSTEM
8x  2 y  4
4(  2 x  3 y)(13)4
8x  2 y  4
 8 x  12 y  52
The coefficient of the x in Equation 1 is 8.
The coefficient of the x in Equation 2 is -2.
To get opposites, multiply Equation 2 by 4.
Now add the equations together
14 y  56
Now substitute 4 for y in either of the two
y4
equations.
8 x  2(4)  4
1 Therefore the solution is (-1/2 , 4).
x
8x  4
2
 1
 2    3(4)  13 Check to see if (-1/2, 4) works in the other equation.
 2
It works! So (-1/2, 4) is the point where the two lines intersect.
EXAMPLE 3: SOLVE THE SYSTEM
2x  3y  2
-3(3 x  y ) (4)-3
2x  3 y  2
 9 x  3 y  12
The coefficient of the y in Equation 1 is -3.
The coefficient of the y in Equation 2 is -1.
To get opposites, multiply Equation 2 by -3.
Now add the equations together
 7 x  14
Now substitute -2 for x in either of the two
x  2
equations.
2(2)  3 y  2 Check to see if (-2, -2) works in the other equation.
3(2)  (2)  4
 4  3y  2
 6  2  4
 3y  6
It works! So (-2, -2) is the point where the two lines intersect.
y  2
Therefore the solution is (-2, -2).
EXAMPLE 4 - BIRDHOUSES
You are selling handmade birdhouses and bird feeders. You sell birdhouses for $12.50
and bird feeders for $15. You earned $245 from selling a total of 17 items. How many
birdhouses and bird feeders did you sell?
Let x = the number of birdhouses you sold
Let y = the number of bird feeders you sold
How many did you sell altogether?
- 15 x  (15) y  255
12.5 x  15 y  245
So.... x  y  17
What was your total earnings?
So.... 12.5 x  15 y  245
-15( x 
y) 17 (-15)
12.5 x  15 y  245
Check:
You sold 4 birdhouses and 13 bird feeders!
 2.5x  10
x4
4  y  17
y  13
12.5(4)  15(13)  245
50  195  245
SOLVE THE SYSTEM
2x  3y  8
 4x  6 y  7
Multiply Top
Equation by 2
4 x  6 y  16
 4x  6 y  7
What happens when you add the two
equations together?
Does that make sense?
0  23
Therefore there is no solution
(The lines are parallel and don’t intersect)
2
The slope of the 1st equation is : m 
3
2
The slope of the 2nd equation is : m 
3
SOLVE THE SYSTEM
5 x  2 y  11
20 x  8 y  44
Multiply Top
Equation by -4
What happens when you add the two
equations together?
Does that make sense?
 20 x  8 y  44
20 x  8 y  44
00
Therefore there is an infinite number of solutions.
(They are the same line.)
ASSIGNMENT
Pages 142-144
Problems 15-39
(multiples of 3)