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Transcript
MAT150
Unit 4-4 -Systems of Equations
Linear and Linear
Objectives
 Solve
 Solve
systems of linear equations graphically
systems of linear equations algebraically with the
substitution method
 Solve systems of linear equations algebraically by
elimination
 Model systems of equations to solve problems
 Determine if a system of linear equations is
inconsistent or dependent
 Solve systems of nonlinear equations algebraically
System of Linear Equations
A system of equations is a collection of one or more
equations, each with one or more variables.
A solution of a system of equations consists of values of
the variables that are solutions to each equation of the
system. To solve a system of equations means to find all
solutions of the system.
Dependent and Inconsistent Systems
Unique solution
Graphs are intersecting lines.
No solution
Graphs are parallel
lines; the system is
inconsistent.
Dependent and Inconsistent Systems
Many solutions Graphs are the same line; the system is
dependent.
Solution by Substitution
One equation is solved for a variable and that variable is
replaced by the equivalent expression in the other
equation.
Example
𝑥 = 4𝑦 + 2
Solve the system by substitution.
6𝑥 − 4𝑦 = 10
Solution
Solution by Elimination
We rewrite one or both of the equations in an equivalent
form that allows us to eliminate one of the variables by
adding or subtracting the equations.
Example
Use the elimination method to solve the system.
3 x  5 y  1

6 x  2y  14
Solution
Modeling Systems of Equations
Solution of real problems sometimes requires us to create
two or more equations whose simultaneous solution is
the solution to the problem.
Example
An investor has $200,000 to invest, part at 8% and the
remainder at 3%. If her investment goal is to have an
annual income of $13,600, how much should she put in
each investment?
Solution
Example (cont.)
An investor has $200,000 to invest, part at 8% and the remainder at 3%. If her investment goal is to
have an annual income of $13,600, how much should she put in each investment?
Solution
Example
Use the elimination method to solve each of the following
systems, if possible. Verify the solution graphically.
a. 2x  5 y  7
b. 4 x  2y  10


4 x  10y  14
2 x  y  3
Solution
a.
Example (cont)
Use the elimination method to solve each of the following
systems, if possible. Verify the solution graphically.
a. 2x  5 y  7
b. 4 x  2y  10


4 x  10y  14
2 x  y  3
Solution
b.
.
Example
2

x
 x  3y  0
Use substitution to solve the system.

 x  y  4
Solution
Example
2

y


x
 2x  8
Solve the system 
2 x  y  4
Solution