Text 7. Solving Linear Systems

ALGEBRA PROJECT UNIT 7 SOLVING SYSTEMS OF LINEAR EQUALITIES

SOLVING SYSTEMS OF LINEAR EQUALITIES AND INEQUALITIES

Lesson 1

Graphing Systems of Equations

Lesson 2

Substitution

Lesson 3

Elimination Using Addition and Subtraction

Lesson 4

Elimination Using Multiplication

Lesson 5

Graphing Systems of Inequalities

GRAPHING SYSTEMS OF EQUATIONS

Example 1

Number of Solutions

Example 2

Solve a System of Equations

Example 3

Write and Solve a System of Equations

Use the graph to determine whether the system has no solution, one solution, or infinitely many solutions.

Answer: Since the graphs of and are parallel, there are no solutions.

Use the graph to determine whether the system has no solution, one solution, or infinitely many solutions.

Answer: Since the graphs of are intersecting lines, there is one solution.

and

Use the graph to determine whether the system has no solution, one solution, or infinitely many solutions.

Answer: Since the graphs of coincide, there are infinitely many solutions.

and

Use the graph to determine whether each system has no solution, one solution, or infinitely many solutions. a. Answer: one b. Answer: no solution c. Answer: infinitely many

Graph the system of equations. Then determine whether the system has no solution, one solution, or infinitely many solutions. If the system has one solution, name it.

Answer: The graphs of the equations coincide. There are infinitely many solutions of this system of equations.

Graph the system of equations. Then determine whether the system has no solution, one solution, or infinitely many solutions. If the system has one solution, name it.

Answer: The graphs of the equations are parallel lines. Since they do not intersect, there are no solutions of this system of equations.

Graph the system of equations. Then determine whether the system has no solution, one solution, or infinitely many solutions. If the system has one solution, name it. a.

Answer: one; (0, 3)

Graph the system of equations. Then determine whether the system has no solution, one solution, or infinitely many solutions. If the system has one solution, name it. b.

Answer: no solution

Bicycling Tyler and Pearl went on a 20-kilometer bike ride that lasted 3 hours. Because there were many steep hills on the bike ride, they had to walk for most of the trip. Their walking speed was 4 kilometers per hour. Their riding speed was 12 kilometers per hour. How much time did they spend walking? Words You have information about the amount of time spent riding and walking. You also know the rates and the total distance traveled. Variables

Let the number of hours they rode and the number of hours they walked. Write a system of equations to represent the situation.

Equations The number of hours riding

r The distance traveled riding

12r

plus

+

the number of hours walking

w

the distance plus traveled walking

+

4w

equals

= equals

=

the total number of hours of the trip.

3 the total distance of the trip.

20

Graph the equations

and

The graphs appear to intersect at the point with the coordinates (1, 2). Check this estimate by replacing r with 1 and w with 2 in each equation.

.

Check

Answer: Tyler and Pearl walked for 3 hours.

Alex and Amber are both saving money for a summer vacation. Alex has already saved $100 and plans to save $25 per week until the trip. Amber has $75 and plans to save $30 per week. In how many weeks will Alex and Amber have the same amount of money? Answer: 5 weeks number of weeks amount of money saved

SUBSTITUTION

Example 1

Solve Using Substitution

Example 2

Solve for One Variable, Then Substitute

Example 3

Dependent System

Example 4

Write and Solve a System of Equations

Use substitution to solve the system of equations.

Since

substitute 4y for x in the second equation. Second equation Simplify. Combine like terms. Divide each side by 15. Simplify.

Use

to find the value of x. First equation Simplify.

Answer: The solution is (20, 5).

Use substitution to solve the system of equations.

Answer: (1, 2)

Use substitution to solve the system of equations.

Solve the first equation for y since the coefficient of y is 1. First equation Subtract 4x from each side. Simplify.

Find the value of x by substituting second equation.

for y in the

Second equation Distributive Property Combine like terms. Add 36 to each side. Simplify. Divide each side by 10. Simplify.

Substitute 5 for x in either equation to find the value of y. First equation Simplify. Subtract 20 from each side. Answer: The solution is (5, –8). The graph verifies the solution.

Use substitution to solve the system of equations.

Answer: (–3, 2)

Use substitution to solve the system of equations.

Solve the second equation for y. Second equation Subtract x from each side. Simplify. Substitute

for y in the first equation. First equation Distributive Property Simplify.

The statement is false. This means there are no solutions of the system of equations. This is true because the slope-intercept form of both equations show that the equations have the same slope, but different y-intercepts. That is, the graphs of the lines are parallel. Answer: no solution

Use substitution to solve the system of equations.

Answer: infinitely many solutions

Gold Gold is alloyed with different metals to make it hard enough to be used in jewelry. The amount of gold present in a gold alloy is measured in 24ths called karats. 24-karat gold is or 100% gold. Similarly, 18- karat gold is or 75% gold. How many ounces of 18-karat gold should be added to an amount of 12-karat gold to make 4 ounces of 14-karat gold?

Let

the number of ounces of 18-karat gold and the number of ounces of 12-karat gold. Use the table to organize the information. Total Ounces

18-karat gold

12-karat gold

14-karat gold

x

y

4

Ounces of Gold

The system of equations is

and

Use substitution to solve this system.

First equation Subtract y from each side. Simplify. Second equation

Distributive Property

Combine like terms. Subtract 3 from each side. Simplify. Multiply each side by –4. Simplify.

First equation

Subtract

from each side.

Simplify. Answer:

ounces of the 18-karat gold and of the 12-karat gold should be used.

ounces

Chemistry Mikhail needs a 10 milliliters of 25% HCl (hydrochloric acid) solution for a chemistry experiment. There is a bottle of 10% HCl solution and a bottle of 40% HCl solution in the lab. How much of each solution should he use to obtain the required amount of 25% HCl solution? Answer: 5mL of 10% solution, 5mL of 40% solution

ELIMINATION USING ADDITION and SUBTRACTION

Example 1

Elimination Using Addition

Example 2

Write and Solve a System of Equations

Example 3

Elimination Using Subtraction

Example 4

Elimination Using Subtraction

Use elimination to solve the system of equations. Since the coefficients of the x terms, –3 and 3, are additive inverses, you can eliminate the x terms by adding the equations. Write the equation in column form and add. Notice that the x value is eliminated. Divide each side by –2. Simplify.

Now substitute –15 for y in either equation to find the value of x. First equation Replace y with –15. Simplify. Add 60 to each side. Simplify. Divide each side by –3. Simplify. Answer: The solution is (–24, –15).

Use elimination to solve the system of equations.

Answer: (2, 1)

Four times one number minus three times another number is 12. Two times the first number added to three times the second number is 6. Find the numbers. Let x represent the first number and y represent the second number. Four times one number

minus

–

4x

Two times the first number

2x

three times another number

added to

+

3y

three times the second number

3y

is

12.

=

12

is

6.

=

6

Use elimination to solve the system. Write the equation in column form and add. Notice that the y value is eliminated. Divide each side by 6. Simplify.

Now substitute 3 for x in either equation to find the value of y. First equation Replace x with 3. Simplify. Subtract 12 from each side. Simplify. Divide each side by –3. Simplify. Answer: The numbers are 3 and 0.

Four times one number added to another number is –10. Three times the first number minus the second number is –11. Find the numbers. Answer: –3, 2

Use elimination to solve the system of equations.

Since the coefficients of the x terms, 4 and 4, are the same, you can eliminate the x terms by subtracting the equations. Write the equation in column form and subtract. Notice that the x value is eliminated. Divide each side by 5. Simplify.

Now substitute 2 for y in either equation to find the value of x. Second equation Simplify. Add 6 to each side. Simplify. Divide each side by 4. Simplify. Answer: The solution is (6, 2).

Use elimination to solve the system of equations.

Answer: The solution is (2, –6).

Multiple-Choice Test Item If and A (3, –8)

what is the value of y? B3 C –8

Read the Test Item You are given a system of equations, and you are asked to find the value of y.

D (–8, 3)

Solve the Test Item You can eliminate the y terms by subtracting one equation from the other. Write the equation in column form and subtract. Notice that the y value is eliminated. Divide each side by 14. Simplify.

Now substitute 3 for x in either equation to solve for y. First equation Simplify. Subtract 24 from each side. Simplify. Notice that B is the value of x and A is the solution of the system of equations. However, the question asks for the value of y. Answer: C

Multiple-Choice Test Item If A4

and

Answer: D

what is the value of x? B (4, –4) C (–4, 4)

D –4

ELIMINATION USING MULTIPLICATION

Example 1

Multiply One Equation to Eliminate

Example 2

Multiply Both Equations to Eliminate

Example 3

Determine the Best Method

Example 4

Write and Solve a System of Equations

Use elimination to solve the system of equations.

Multiply the first equation by –2 so the coefficients of the y terms are additive inverses. Then add the equations. Multiply by –2. Add the equations. Divide each side by –1. Simplify.

Now substitute 9 for x in either equation to find the value of y. First equation Simplify. Subtract 18 from each side. Simplify. Answer: The solution is (9, 5).

Use elimination to solve the system of equations.

Answer: (5, 1)

Use elimination to solve the system of equations.

Method 1 Eliminate x. Multiply by 3. Multiply by –4. Add the equations. Divide each side by 29. Simplify.

Now substitute 4 for y in either equation to find x. First equation Simplify. Subtract 12 from each side. Simplify. Divide each side by 4. Simplify. Answer: The solution is (–1, 4).

Method 2 Eliminate y. Multiply by 5. Multiply by 3. Add the equations. Divide each side by 29. Simplify.

Now substitute –1 for x in either equation. First equation Simplify. Add 4 to each side. Simplify. Divide each side by 3. Simplify. Answer: The solution is (–1, 4), which matches the result obtained with Method 1.

Use elimination to solve the system of equations.

Answer: (4, –1)

Determine the best method to solve the system of equations. Then solve the system.

•For an exact solution, an algebraic method is best. •Since neither the coefficients for x nor the coefficients for y are the same or additive inverses, you cannot use elimination using addition or subtraction. •Since the coefficient of the x term in the first equation is 1, you can use the substitution method. You could also use the elimination method using multiplication.

The following solution uses substitution. First equation Subtract 5y from each side. Simplify.

Second equation

Distributive Property Combine like terms. Subtract 12 from each side. Simplify. Divide each side by –22. Simplify.

First equation

Simplify. Subtract 5 from each side. Simplify. Answer: The solution is (–1, 1).

Determine the best method to solve the system of equations. Then solve the system.

Answer: The best method to use is elimination using subtraction because the coefficient of y is the same in both equations; (3, 5).

Transportation A fishing boat travels 10 miles downstream in 30 minutes. The return trip takes the boat 40 minutes. Find the rate of the boat in still water. Let the rate of the boat in still water. Let the rate of the current. Use the formula rate × time distance, or Since the rate is miles per hour, write 30 minutes as hour and 40 minutes as hour.

r

t

d

Downstream

10

Upstream

10

This system cannot easily be solved using substitution. It cannot be solved by just adding or subtracting the equations. The best way to solve this system is to use elimination using multiplication. Since the problem asks for b, eliminate c.

Multiply by

.

Multiply by

.

Add the equations. Multiply each side by Simplify. Answer: The rate of the boat is 17.5 mph.

Transportation A helicopter travels 360 miles with the wind in 3 hours. Te return trip against the wind takes the helicopter 4 hours. Find the rate of the helicopter in still air. Answer: 102.5 mph

GRAPHING SYSTEMS OF INEQUALITIES

Example 1

Solve by Graphing

Example 2

No Solution

Example 3

Use a System of Inequalities to Solve a Problem

Example 4

Use a System of Inequalities

Solve the system of inequalities by graphing. Answer: The solution includes the ordered pairs in the intersection of the graphs of and The region is shaded in green. The graphs and are boundaries of this region. The graph is dashed and is not included in the graph of . The graph of is included in the graph of

Solve the system of inequalities by graphing.

Answer:

Solve the system of inequalities by graphing.

Answer: The graphs of

and are parallel lines. Because the two regions have no points in common, the system of inequalities has no solution.

Solve the system of inequalities by graphing.

Answer: Ø

Service A college service organization requires that its members maintain at least a 3.0 grade point average, and volunteer at least 10 hours a week. Graph these requirements. Words The grade point average is at least 3.0. The number of volunteer hours is at least 10 hours. Variables If the grade point average and the number of volunteer hours, the following inequalities represent the requirements of the service organization.

Inequalities The grade point average is at least 3.0. The number of volunteer hours is at least 10. Answer: The solution is the set of all ordered pairs whose graphs are in the intersection of the graphs of these inequalities.

The senior class is sponsoring a blood drive. Anyone who wishes to give blood must be at least 17 years old and weigh at least 110 pounds. Graph these requirements. Answer:

Employment Jamil mows grass after school but his job only pays $3 an hour. He has been offered another job as a library assistant for $6 per hour. Because of school, his parents allow him to work 15 hours per week. How many hours can Jamil mow grass and work in the library and still make at least $60 per week? Let

the number of hours spent mowing grass and the number of hours spent working in the library. Since g and both represent a number of days, neither can be a negative number. The following system of inequalities can be used to represent the conditions of this problem.

The solution is the set of all ordered pairs whose graphs are in the intersection of the graphs of these inequalities. Only the portion of the region in the first quadrant is used since and . Answer: Any point in the region is a possible solution. For example (2, 10) is a point in the region. Jamil could mow grass for 2 hours and work in the library for 10 hours during the week.

Emily works no more than 20 hours per week at two jobs. Her baby-sitting job pays $3 an hour and her job as a cashier at the bookstore pays $5 per hour. How many hours can Emily work at each job to earn at least $80 per week? Answer: number of hours baby sitting number of hours working as a cashier

THIS IS THE END OF THE SESSION

BYE!