Download Solving Linear Systems by Linear Combinations

7.3
Solving Linear Systems by
Linear Combinations
Goal
Solve a system of linear
equations by linear
combinations.
How can a farmer find the location of a beehive?
In Exercise 44 you will solve a
linear system to find the location
of a beehive. You will use a method
called linear combinations.
Key Words
• linear combination
Sometimes it is not easy to isolate one of the variables in a linear system. In that
case it may be easier to solve the system by linear combinations. A linear
combination of two equations is an equation obtained by (1) multiplying one or
both equations by a constant if necessary and (2) adding the resulting equations.
EXAMPLE
1
Add the Equations
4x 3y 16
2x 3y 8
Solve the linear system.
Equation 1
Equation 2
Solution
Add the equations to get an equation in one variable.
4x 3y 16
Write Equation 1.
2x 3y 8
Write Equation 2.
24
6x
x4
Add equations.
Solve for x.
Substitute 4 for x into either equation and solve for y.
4(4) 3y 16
y0
Substitute 4 for x.
Solve for y.
Check by substituting 4 for x and 0 for y in each of the original equations.
ANSWER 䊳
The solution is (4, 0).
Add the Equations
Solve the linear system. Then check your solution.
1.
402
Chapter 7
3x 2y 7
3x 4y 5
2. 4x 2y 2
Systems of Linear Equations and Inequalities
3x 2y 12
3.
5x 2y 4
5x 3y 19