systems of equations & inequalities

Systems of Equations & Inequalities

Algebra I

Systems of Equations

Definition Ways to solve means two or more

linear equations

If these two linear equations intersect, that point of intersection is called the solution 

Graphing By hand On calculator

Substitution

Elimination

Graphing Method

This method of solving equations is by graphing each equation on a coordinate graph.

The coordinates of the intersection will be the solution to the system.

Graphing System of linear equations with calculator:the following two lines is a system:

y=x+1y=2x

What is the solution?The point 

(1,2) is where the

two lines intersect.

Practice Problem:Use Calculator:The following two lines is a system:

y = 2x+1 y = 4x - 1What is the solution?

The solution of this system is the point of

intersection : (1,3)

X = 4 + y x – 3y = 4

~ Find three values for x and y that satisfy each equation.x = 4 + y x – 3y = 4

~ Graph these points and draw straight lines

The point where the two lines cross (4, 0) is the solution of the system.

Graphing System of linear equations by hand ~ Using Table:

2y = 4x + 2 2y = -x + 7

~ Get the equations in slope intercept form , so you have y-intercept and slope to graph line:2y = 4x + 2

y = (4x + 2) / 2y = 2x +1

2y = 8x - 2y = (8x - 2) / 2y = 4x – 1

The point where the two lines cross (1, 3) is the solution of the system.

Solve the following system of linear equations graphing the system by hand using slope intercept form:

Graphing and getting Parallel Lines If the lines are parallel

they do not intersect there is no solution to that system.

Substitution Method

Sometimes a system is more easily solved by the substitution method. This method involves substituting

one equation into another.

x = y + 8 x + 3y = 48If an equation is not already solved for one variable then you need to solve for either x or y in order to substitute.

From the first equation, substitute ( y + 8 ) for x in the second equation.

(y + 8) + 3y = 48

Now solve for y. Simplify by combining y's

y + 8 + 3y = 48 4y + 8 = 48 4y = 48 – 8 4y = 40 y = 40/4 y = 10

Now insert y = 10 in one of the original equations.

x = y + 8x = 10 + 8x = 18

Solution:x = 18 , y = 10(18, 10)

y = 2x + 12y = 3x - 2If an equation is not already solved for one variable then you need to solve for either x or y in order to substitute.

From the first equation, substitute ( 2x + 1 ) for y in the second equation.

2(2x + 1) = 3x - 2

Now solve for x:

4x + 2 = 3x - 2 4x – 3x + 2 = - 2 x + 2 = - 2 x = - 2 – 2 x = - 4

Now insert x = - 4 in one of the original equations.

y = 2x + 1y = 2(-4) + 1y = -8 + 1y = -7

Solution:x = -4 , y = -7(-4, -7)

x + y = 11 3x - y = 5Solve the first equation for either x or yNeed one variable equaling an expression so that you can substitute

From the first equation, for y: x + y = 11 y = 11 – x

Substitute 11 - x for y in the second equation:

3x – (11 – x) = 5

Now solve for x. Simplify by combining x‘s:

3x - 11 + x = 5 4x = 5 + 11 4x = 16 x = 16 / 4 x = 4

Substitute 4 for x in either equation and solve for y:

4 + y = 11y = 11 - 4 y = 7

Solution:x = 4 , y = 7(4, 7)

2x – 3y = 6x + y = -12Solve the first equation for either x or yNeed one variable equaling an expression so that you can substitute

From the second equation, for y:

x + y = -12 y = -12 – x

Substitute -12 - x for y in the second equation:

2x – 3(-12 – x) = 6

Now solve for x: 2x + 36 + 3x = 6 5x + 36 = 6 5x = 6 - 36 5x = - 30 x = - 30 / 5 x = - 6

Substitute – 6 for x in either equation and solve for y:

- 6 + y = -12y = -12 + 6 y = - 6

Solution:x = - 6 , y = - 6(-6, -6)

Individual Practice Problems: Substitution Method Problem 1:

y= x + 1 2y= 3x

Problem 2: y – 5x = - 1 2y= 3x + 12

Problem 3: y = 3x + 1 4y = 12x + 4

Problem 4: y – 3x = 1 4y = 12x + 3

Answers to Practice Problems Problem 1:

y= x + 1 2y= 3x

Answers to Practice Problems Problem 2: y – 5x = - 1 2y= 3x + 12

Solve 1st equation for y: y = 5x - 1

Answers to Practice Problems Problem 3:

y = 3x + 1 4y = 12x + 4

Answers to Practice Problems Problem 4: y – 3x = 1 4y = 12x + 3

Solve 1st equation for y: y = 3x + 1

Elimination MethodHelps eliminate one variable so that you can solve

for the remaining variable.

Steps for Elimination Method Multiply one or both equations by some

number to make the number in front of one of the letters (unknowns) the same in each equation.

Add or subtract the two equations to eliminate one letter.

Solve for the other unknown. Insert the value of the first unknown in

one of the original equations to solve for the second unknown.

x + y = 7x – y = 3~ You have opposites between the two equations a positive and negative y

~ With opposites you can skip step one and add equations together

x + y = 7 + x – y = 3 2x = 10

solve for x: 2x = 10 x = 10 / 2 x = 5

replace x value and find y: 5 + y = 7 y = 7 – 5 y = 2

Solution:x = 5 and y = 2(5, 2)

3x + 3y = 242x + y = 13~ You need to have the same variable in both equations…

~ With the same variable you can subtract equations together

Need to get the same y’s in both:

change 2nd equation to have 3y

3(2x + y) = 3(13)

6x + 3y = 39 3x + 3y =

24 - 6x + 2y =

39 -3x = -

15

solve for x:

-3x = -15 x = -15 / -3 x = 5

replace x value and find y: 2x + y = 13 2(5) + y = 13 10 + y = 13 y = 13 - 10 y = 3

Solution:x = 5 and y = 2(5, 2)

5x + 3y = 7 3x - 5y = -23~ You need to have the same variable in both equations…

~ With the same variable you can subtract equations together

Multiply the second equation by 5 to make the x-coefficient a multiple of 5:

5(3x - 5y) = 5(-23) 15x - 25y = -115

multiply the first equation by 3, to get the same x-coefficient:

3(5x + 3y) = 3(7) 15x + 9y = 21

15x - 25y = -115 -15x + 9y = 21

-34y = -136

solve for y: -34y = -136 y = -136 / -34 y = 4

replace 4 for y in one of the original equations and find x: 5x + 3y = 7 5x + 3(4)= 7 5x + 12 = 7 5x = 7 - 12 5x = -5 x = -5 / 5 x = -1

Solution:x = -1 & y = 4(-1, 4)

Individual Practice Problems: Elimination Method Problem 1:

y = x + 1  y = –x

Problem 2: 4x – 2y = 14 x + 2y = 6

Problem 3: 2x + 3y = 17 4x – 3y = 1

Problem 4: y = 2x + 1 y = -4x + 1

Answers to Practice Problems Problem 1:

y = x + 1  y = –x

Answers to Practice Problems Problem 2:

4x – 2y = 14 x + 2y = 6

Answers to Practice Problems Problem 3:

2x + 3y = 17 4x – 3y = 1

Answers to Practice Problems Problem 4:

y = 2x + 1 y = -4x + 1

Need to get same coefficient fro the x variables

Story Problemswith Systems of Equations

Steps to Solve Define the variables Set up the equations Solve the system

Graphing Substitution Elimination

Put answer back into form of the problem

The admission fee at a small fair is $1.50 for children and $4.00 for adults. On a certain day, 2200 people enter the fair and $5050 is collected. How many children and how many adults attended? Define Variables: number of adults: a

number of children: c Write Equations: total number: 

a + c = 2200 total income: 4a + 1.5c = 5050

Solve for variables: Solve 1st equation for a: a = 2200 – c Use (2200 – c) for a in

2nd to find c:

4(2200 – c) + 1.5c = 5050

8800 – 4c + 1.5c =

5050  8800 – 2.5c = 5050  -2.5c = 5050 - 8800 –2.5c = –3750  c = -3750 / -2.5 c = 1500 Use 1500 for c to find a: a = 2200 – (1500) =

700 Write out answer There were 1500 children

and 700 adults.

A landscaping company placed two orders with a nursery. The first order was for 13bushes and 4 trees, and totaled $487. The second order was for 6 bushes and 2 trees, and totaled $232. The bills do not list the per-item price. What were the costs of one bush and of one tree?

Define Variables: b = the number of bushes t = the number of trees Write Equations: first order: 13b + 4t = 487 

second order: 6b + 2t = 232 Solve for variables: use elimination

Multiplying the second by –2 -2 (6b + 2t) = -2(232) -12b – 4t = -464

13b + 4t = 487 + –12b – 4t = –464

b = 23

Use 23 for b to find t: 13b + 4t = 487 13(23) + 4t = 487 299 + 4t = 487 4t = 487 – 299 4t = 188 t = 188 / 4 t = 47 Write out answer Bushes

cost $23 each; trees cost $47 each.

The sum of the digits of a two-digit number is 7. When the digits are reversed, the number is increased by 27. Find the number. Define Variables: t = tens digit of the original number u = units (or "ones") digit Write Equations: 1st Sentence gives you:

t + u = 7 Think about numbers to get original numbers:

26 is 10 times 2, plus 6 times 1 two-digit number will be ten times the (tens digit), plus one

times the (units digit) original number: 10t + 1u New number has the digits reversed:(switch the place of t & u)

new number: 10u + 1t (new number) is (old number) increased by (twenty-seven) 10u + 1t = 10t + 1u + 27

The sum of the digits of a two-digit number is 7. When the digits are reversed, the number is increased by 27. Find the number. ~ Continued

System to solve: t + u = 7  10u + t = 10t + u + 27

Simplify the second equation: 10u + t = 10t + u + 27 10u – u + t – 10t = 27 9u – 9t = 27  9(u – t) = 27 u – t = 27 / 9 u – t = 3

Write to look like other: -t + u = 3

Solve for variables: Use elimination! t + u = 7 + -t + u = 3 2u = 10 u = 10 / 2 u = 5 Use 5 for u to find t: t + u = 7 t + 5 = 7 t = 7 – 5 t = 2 Write out answer The number is 25.

A passenger jet took three hours to fly 1800 miles in the direction of the jetstream. The return trip against the jetstream took four hours. What was the jet's speed in still air and the jetstream's speed?

Define Variables:  p = the plane's speedometer reading w = the windspeed Write Equations:

the plane is going "with" the wind the two speeds will be added together when the plane is going "against" the wind the windspeed will be subtracted from

the plane's speedometer reading the distance equation will be:

(the combined speed) times (the time at that speed) equals (the total distance travelled)

with the jetstream: against the jetstream:  (p + w)(3) = 1800 (p – w)(4) = 1800 p + w = 1800 / 3 p – w = 1800 / 4 p + w = 600 p – w = 450 

A passenger jet took three hours to fly 1800 miles in the direction of the jetstream. The return trip against the jetstream took four hours. What was the jet's speed in still air and the jetstream's speed?

Solve for variables: use elimination Method p + w = 600 

+ p – w = 450 2p = 1050 p = 1050 / 2 p = 525

Use 525 for p to find w: p + w = 600 525 + w = 600

w = 600 – 525 w = 75

Write out answer The jet's speed

was 525 mph and the jetstream windspeed was 75 mph.

Systems of Inequalities

Solving Systems of Linear Inequalities

The solution to a system of linear inequalities is shown by graphing them.

Need to put the inequalities into Slope-Intercept Form, y = mx + b.

Solving Systems of Linear Inequalities

Lines on the graph

o If the inequality is < or >, make the lines dotted.

o If the inequality is < or >, make the lines solid.

Solving Systems of Linear Inequalities

The solution also includes points not on the line, so you need to shade the region of the graph:

above the line for ‘y >’ or ‘y ’. below the line for ‘y <’ or ‘y ≤’.

Solving Systems of Linear Inequalities

Example: a: 3x + 4y > - 4

b: x + 2y < 2

Put in Slope-Intercept Form:

) 3 4 44 3 4

3 14

a x yy x

y x

) 2 22 2

1 12

b x yy x

y x

Solving Systems of Linear Inequalities

a: dotted

shade above

b:dottedshade below

Graph each line, make dotted or solid and shade the correct area.

Example, continued:

3: 14

a y x 1: 12

b y x

Solving Systems of Linear Inequalities

a: 3x + 4y > - 4

3: 14

a y x

Solving Systems of Linear Inequalities

a: 3x + 4y > - 4

b: x + 2y < 2

3: 14

a y x

1: 12

b y x

Solving Systems of Linear Inequalities

a: 3x + 4y > - 4 b: x + 2y < 2

The area between the green arrows is the region of overlap and thus the solution.