Ch 10.1 : Solving systems by Substitutions and
Elimination
In this section, we will
1. define and classify linear systems of equations
2. solve linear systems by substitution
3. solve linear systems by elimination
4. consider a larger systems of equations
5. look at some applications of systems of equations
Definition of a linear system
A linear system is a collection of linear equations.
I A linear system of equations with no solutions is said to be
inconsistent.
I A linear system of equations that has at least one solution is
called consistent.
Remark: A system of two linear equations then can have
1. 0 solution
2. 1 solution
3. infinitely many solutions
Solving systems by substitution
Given a linear system of equation, solve for one variable, and
substitute the result for the variable in the remaining equations.
Example) Use the method of substitution to solve the system
2x − y = 1
x +y =5
Example
Use the method of substitution to solve the system
3x + y = 4
−2x + 3y = 1
Solving systems by Elimination
Eliminate one variable in one equation by adding two equations
(may not the equations that you start out with) together.
Example) Use the method of elimination to solve the system
5x + 3y = −7
7x − 6y = −20
Example
Use the method of elimination to solve the system
2x − 3y = 3
9
3x − y = 11
2
Lager systems of equations (3 equations and 3 unknowns)
Solve the system
3x − 5y + z = −10
−x + 2y − 3z = −7
x − y − 5z = −24
Example
Solve the system
x − 4y + 2z = −1
2x + y − 3z = 10
−3x + 12y − 6z = 3
Example
Solve the system
x + 2y = −1
y + 3z = 7
2x + 5z = 21
Applications of systems of equations
A foundry needs to produce 75 tons of an alloy that is 34% copper.
It has supplies of 9% copper alloy and 84% copper alloy. How
many tons of each must be mixed to obtain the desired result?
Homework for Ch 10.1
Show all work to get credit
1, 5, 7, 8, 14, 17, 18, 22, 41, 44, 56, 73, 76, 77, 79, 102