Download Chapter 4 Section 4.1: Solving Systems of Linear Equations by

Chapter 4
Section 4.1: Solving Systems of Linear Equations by Graphing
Objectives:
1. Decide whether a given ordered pair is a solution of a system.
2. Solve linear systems by graphing.
3. Solve special systems by graphing.
4. Identify special systems without graphing.
Deciding Whether a Given Ordered Pair is a Solution
A system of linear equations, often called a linear system, consists of two or more linear equations with
the same variables.
A solution of a system of linear equations is an ordered pair that makes both equations true at the same
time. A solution of an equation is said to satisfy the equation.
Example: Is (2,–1) a solution of the system
3x + y = 5
2x – 3y = 7 ?
Example: Is (2,–1) a solution of the system
x + 5y = –3
4x + 2y = 1 ?
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Solving Linear Systems by Graphing
Example 2
Solve the system of equations by graphing both equations on the same axes.
–2x + ⅔y = –4
5x – y = 8
Step1: Rewrite each equation in slope-intercept form to graph.
Step 2: Graph both lines on the same axes and identify where they cross.
CAUTION
A difficulty with the graphing method of solution is that it may not be possible to determine from the
graph the exact coordinates of the point that represents the solution, particularly if these coordinates
are not integers. For this reason, algebraic methods of solution are explained later in this chapter. The
graphing method does, however, show geometrically how solutions are found and is useful when
approximate answers will do.
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Solving Special Systems by Graphing
Solve each system by graphing.
(a) 3x + y = 4
6x + 2y = 1
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(b) ½x + y = 3
2x + 4y = 12
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Three Cases for Solutions of Systems
1. The graphs intersect at exactly one point, which gives the (single) ordered-pair solution of the
system. The system is consistent, and the equations are independent.
2. The graphs are parallel lines. So, there is no solution and the solution set is Ø. The system is
inconsistent and the equations are independent.
3. The graphs are the same line. There is an infinite number of solutions, and the solution set is
written in set-builder notation. The system is consistentand the equations are dependent.
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Section 4.2: Solving Systems of Linear Equations by Substitution
Objectives:
1. Solve linear systems by substitution.
2. Solve special systems by substitution.
3. Solve linear systems with fractions and decimals by substitution.
Solving Linear Systems by Substitution
Step 1 Solve one equation for either variable. If one of the variables has a coefficient of 1 or – 1,
choose it, since it usually makes the substitution easier.
Step 2 Substitute for that variable in the other equation. The result should be an equation with just one
variable.
Step 3 Solve the equation from Step 2.
Step 4 Substitute the result from Step 3 into the equation from Step 1 to find the value of the other
variable.
Step 5 Check the solution in both of the original equations. Then write the solution set.
Example: Solve the system by the substitution method.
The second equation is already solved for y. This equation says that y = – 3x. Substituting – 3x
for y in the first equation gives
Because x = – 2, we find y from the equation y = – 3x by substituting – 2 for x.
Check that the solution of the given system is (– 2, 6) by substituting – 2 for x and 6 for y in both
equations.
The solution set of the system is {(– 2, 6)}.
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Example: Solve the system by the substitution method.
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Example: Solve the system by the substitution method.
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Example: Solve the system by the substitution method.
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Example: Solve the system by the substitution method.
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Section 4.3: Solving Systems of Linear Equations by Elimination
Objectives:
1. Solve linear systems by elimination.
2. Multiply when using the elimination method.
3. Use an alternative method to find the second value in a solution.
4. Use the elimination method to solve special systems.
Solving Linear Systems by Elimination
Using the addition property to solve systems is called the elimination method. When using this
method, the idea is to eliminate one of the variables. To do this, one of the variables in the two
equations must have coefficients that are opposites.
Solving a Linear System by Elimination
Step 1 Write both equations in standard form Ax + By = C.
Step 2 Transform so that the coefficients of one pair of variable terms are opposites. Multiply one or
both equations by appropriate numbers so that the sum of the coefficients of either the x- or y-terms is
0.
Step 3 Add the new equations to eliminate a variable. The sum should be an equation with just one
variable.
Step 4 Solve the equation from Step 3 for the remaining variable.
Step 5 Substitute the result from Step 4 into either of the original equations and solve for the other
variable.
Step 6 Check the solution in both of the original equations. Then write the solution set.
Example: Use the elimination method to solve the system.
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CAUTION
A system is not completely solved until values for both x and y are found. Do not stop after finding the
value of only one variable. Remember to write the solution set as a set containing an ordered pair.
Example: Use the elimination method to solve the system.
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Example: Use the elimination method to solve the system.
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Example: Use the elimination method to solve the system.
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Section 4.4: Applications of Linear Systems
Objectives:
1. Solve problems about unknown numbers.
2. Solve problems about quantities and their costs.
3. Solve problems about mixtures.
4. Solve problems about distance, rate (or speed), and time.
Solving an Applied Problem with Two Variables
Step 1 Read the problem, several times if necessary, until you understand what is given and what is to
be found.
Step 2 Assign variables to represent the unknown values, using diagrams or tables as needed. Write
down what each variable represents.
Step 3 Write two equations using both variables.
Step 4 Solve the system of two equations.
Step 5 State the answer to the problem. Is the answer reasonable?
Step 6 Check the answer in the words of the original problem.
Example: Amy and Bryan just celebrated a birthday together. Amy is 3 years older than Bryan.
However, if Bryan doubled his age and subtracted 5, the result would be 4 years older than Amy. How
old are Amy and Bryan?
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Example: Carrie and Diego are headed to the vending machine and have decided to pool their money.
Carrie has only quarters, Diego has only dimes, and together they have ten coins worth $2.05. How
many quarters does Carrie have, and how many dimes does Diego have?
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Example: A pharmacist needs 100L of 50% alcohol solution. She has on hand 30% alcohol solution and
80% alcohol solution, which she can mix. How many liters of each will be required to make the 100L of
50% alcohol solution?
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Example: A plane flies 560 mi in 1.75 hr traveling with the wind. The return trip against the same wind
takes the plane 2hr. Find the speed of the plane and the speed of the wind.
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Section 4.5: Solving Systems of Linear Inequalities
Objectives:
1. Solve systems of linear inequalities by graphing.
Solving a System of Linear Inequalities
Step 1 Graph the inequalities. Graph each inequality using the method of Section 3.5.
Step 2 Choose the intersection. Indicate the solution set of the system by shading the intersection of
the graphs (the region where the graphs overlap).
Example: Graph the solution set of the system.
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Example: Graph the solution set of the system.
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Example: Graph the solution set of the system.
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