4.1 Solving Systems of Linear Equations in Two Variables
Welcome message from author
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
4.1
Solving Systems of Linear Equations in
Two Variables
Systems of Linear Equations
A system of equations consists of two or more equations.
The solution of a system of two equations in two variables is an ordered pair (x, y) that makes both equations true.
Determine whether (–3, 1) is a solution of the system.
x – y = – 4
2x + 10y = 4
Example
Determine whether (4, 2) is a solution of the system.
2x – 5y = – 2
3x + 4y = 4
Example
Since a solution of a system of equations is a solution common to both equations, it is also a point common to the graphs of both equations.
To find the solution of a system of two linear equations, we graph the equations and see where the lines intersect.
Solving Systems of Equations by Graphing
Solve the system of equations by graphing.
Example
53
23
2
xy
xy
Solve the system of equations by graphing.
2x – y = 6x + 3y = 10
Example
Solve the system of equations by graphing.
–x + 3y = 6
3x – 9y = 9
continued
Example
Solve the system of equations by graphing.x = 3y – 1 2x – 6y = –2
Example
There are three possible outcomes when graphing two linear equations in a plane.
One point of intersection—one solution
Parallel lines—no solution
Coincident lines—infinite number of solutions
If there is at least one solution, the system is considered to be consistent.
If the system defines distinct lines, the equations are independent.
Identifying Special Systems of Linear Equations
Possible Solutions of Linear Equations
Consistent
The equations are dependent.
If the lines lie on top of each other, then the system has infinitely many solutions. The solution set is the set of all points on the line.
Inconsistent
The equations are independent.
If the lines are parallel, then the system of equations has no solution because the lines never intersect.
Consistent
The equations are independent.
If the lines intersect, the system of equations has one solution given by the point of intersection.
Graph Type of SystemNumber of Solutions
Two lines intersect at one point.
Parallel lines
Lines coincide
(3, 5)
The Substitution Method
Another method that can be used to solve systems of equations is called the substitution method.
To use the substitution method, we first need an equation solved for one of its variables. Then substitute that new expression for the variable into the other equation and solve for the other variable.
Solve the system using the substitution method.
6x – 4y = 10Y = 3x - 3
Example
continued
Solve the system using the substitution method.
3x – y = 14x + y = 6
Example
continued
Solve the system using the substitution method.
3x – y = 6– 4x + 2y = –8
Example
continued
Solving a System of Two Equations Using the Substitution Method
Step 1: Solve one of the equations for one of its variables.Step 2: Substitute the expression for the variable found in
Step 1 into the other equation.Step 3: Find the value of one variable by solving the
equation from Step 2.Step 4: Find the value of the other variable by substituting
the value found in Step 3 into the equation from Step 1.
Step 5: Check the ordered pair solution in both original equations.
The Substitution Method
Solve the system:
y = 2x – 5
8x – 4y = 20
Example
continued
Solve the following system of equations:
3x – y = 46x – 2y = 4
Example
continued
Another method that can be used to solve systems of equations is called the addition or elimination method.
You multiply both equations by numbers that will allow you to combine the two equations and eliminate one of the variables.
Solving a System Using Elimination
Solve the following system
Example
x + y = 7x – y = 9
x – 5y = -12-x + y = 4
Solve the following system of equations
2x – y = 93x + 4y = –14
Example
continued
Solve the following system of equations
6x – 3y = –34x + 5y = –9
Example
continued
Solving a System of Two Linear Equations Using the Elimination Method
Step 1: Rewrite each equation in standard form, Ax + By = C.
Step 2: If necessary, multiply one or both equations by some nonzero number so that the coefficients of a variable are opposites of each other.
Step 3: Add the equations.
Step 4: Find the value of one variable by solving the equation from Step 3.
Step 5: Find the value of the second variable by substituting the value found in Step 4 into either of the original equations.
Step 6: Check the proposed solution in both original equations.
The Elimination Method
Solve the system of equations using the elimination method.
24
1
2
12
3
4
1
3
2
yx
yx
Example
continued
Related Documents
