Transcript 5.1 Solving Systems of Linear Equations by Graphing • To solve by graphing, graph both linear equations. This gives an approximate solution.

5.1 Solving Systems of Linear
Equations by Graphing
• To solve by graphing, graph both linear equations.
This gives an approximate solution. Algebraic
methods are more exact (next 2 sections).
• If the graphs intersect at one point the system is
consistent and the equations are independent.
5.1 Solving Systems of Linear
Equations by Graphing
• If the graphs are parallel lines, there is no solution and the
solution set is . The system is inconsistent.
• If the graphs represent the same line, there are an infinite
number of solutions. The equations are dependent.
5.2 Solving Systems of Linear
Equations by Substitution
•
Solving by substitution:
1. Solve for a variable
2. Substitute for that variable in the other
equation
3. Solve this equation for the remaining variable
4. Put your solution back into either of the
original equations to solve for the other
variable
5. Check your solution with the other equation
5.2 Solving Systems of Linear
Equations by Substitution
• Example:
2x  y  7
3x  y  13
From the first equation we get y=2x-7, so
substituting into the second equation:
3x  2 x  7  13
5 x  7  13  5 x  20  x  4
24  y  7  8  y  7
 y  1  y  1
5.2 Solving Systems of Linear
Equations by Substitution
• If when using substitution both variables drop out
and you get something like: 10=6
The system inconsistent and there is no solution
(parallel lines)
• If when using substitution both variables drop out
and you get something like: 10=10
The system dependent and every solution of one
line is also on the other (same lines)
5.3 Solving Systems of Linear
Equations by the Addition Method
•
Solving systems of equations by the addition
method (a.k.a. elimination):
1. Write equations in standard form (variables line up)
2. Multiply one of the equations to get coefficients of
one of the variables to be opposites
3. Add (or subtract) equations – so that one variable
drops out
4. Solve for the remaining variable.
5. Plug you solution back into one of the original
equations and solve for the other variable.
5.3 Solving Systems of Linear
Equations by the Addition Method
• Example:
2x  3y  5
4 x  y  17
• Multiply the second equation by 3 to get:
2x  3 y  5
12x  3 y  51
• Adding equations you get:
14x  56  x  4
4  4  y  17  y  17  16  1
5.3 Solving Systems of Linear
Equations by the Addition Method
• If when using elimination both variables drop out
and you get something like: 10=6
The system is inconsistent and there is no solution
(parallel lines)
• If when using elimination both variables drop out
and you get something like: 10=10
The system is dependent and every solution of one
line is also on the other (same lines)
5.4 Applications of Linear
Systems of Equations
•
Solving an applied problem by writing a system
of equations:
1. Determine what you are to find – assign variables
2. Draw a diagram, figure or make a chart of
information.
3. Write the system of equations
4. Solve the system using substitution or elimination
5. Answer the question from the problem.
5.4 Applications of Linear
Systems of Equations
• Mixture problem: How many ounces of a 5%
solution must be added to a 20% solution to get 10
ounces of 12.5% solution.
Let x = # ounces of 5% solution
Let y = # ounces of 20% solution
5.4 Applications of Linear
Systems of Equations
• Solution to mixture problem in 2 variables:
x  y  10
5% x  20% y  12.5%  10
.05x  .210  x   .12510
.05x  2.0  .2 x  1.25
 .15x  1.25  2.0  .75
x  5  y  10  5  5