CHAPTER 7

Systems of Equations & Inequalities

7.1 Systems of Linear Equations in 2 Variables

• Objectives– Decide whether an ordered pair is a solutions

of a linear system.– Solve linear systems by substitution– Solve linear systems by addition.– Identify systems that do not have exactly one

ordered-pair solution.– Solve problems using systems of linear

equations.

What is a solution of a system of linear equations?

• Linear equations have graphs that are lines and are stated in terms of 2 variables, x & y.

• A system of equations indicates there are 2 or more equations, thus 2 or more lines.

• A solution to a system of equations would be the ordered pair that makes both equations true simultaneously.

• Graphically, this would indicate the point of intersection of the lines. (0 points if parallel, 1 point if non-parallel & distinct, or infinitely many if the 2 lines are on top of each other)

Solving a system of 2 equations• Substitution: isolate a variable (x or y) in

one equation & substitute the expression it is equal to into the other equation

• Example: – Solve: x – 3y = -11 and x + y = 5– Add 3y to both sides of the 1st equation,

resulting in: x = 3y – 11– Now replace x in the 2nd equation with this:

(3y – 11) + y = 5 OR 4y = 16 OR y = 4

NOW solve for x in EITHER equation (let y = 4)

x + 4 = 5 OR x = 1 SOLUTION: (1,4)

CHECK the solution in both equations…is it true?

Addition to solve a system of equations

• Add (or subtract) the 2 equations to eliminate one variable, then solve for the other

• Add the 2 equations together:• Example: Solve: 2x – y = 2, 3x – y = 5

• Let x = 3 in either equation to solve for y:• 2(3) – y = 2, y = 4 • Solution: (3,4) (CHECK!)

3

,155

133

22

x

x

yx

yx

Sometimes, you must multiply an equation by a constant to get a term to drop out when adding

• Example:

• Solve: 2x – 3y = 7, 4x + 5y = 25

• Note: If the 2 equations were added, nothing would drop out.

• Multiply 1st equation (both sides) by (-2)

• -4x + 6y = -14 (NOW add to 2nd eq.)

• Result: 11y = 11, thus y = 1

• Solve for x: 2x – 3 = 7, x = 5

• Solution: (5,1) CHECK!!

Systems with NO solution

• Since the equations are linear, no solutions would mean the lines are parallel.

• How would we know this w/o graphing?

• When solving (either w/substitution or graphing) we would encounter a FALSE statement. (example, next slide)

System with NO solution

• Solve : 3x + y = 5, -6x – 2y = 12

• Substitute: let y = 5 – 3x

• -6x – 2(5 – 3x) = 12

• -6x – 10 + 6x = 12

• -10 = 12 FALSE!!! No matter what values you use for (x,y), this statement will never be true:

• NO SOLUTION (lines are parallel)

Systems with infinitely many solutions

• Lines would lie on top of each other • Example: 2x + 4y = 6, x + 2y = 3• Multiply 2nd equation by (-2): -2x – 4y = -6• Add the 2 equations together:• 0 + 0 = 0 When is this true??? ALWAYS• There are infinitely many solutions. They are all

ordered pairs that fall on that line.• Possible pairs: (3,0), (5,-1), (-1,2), etc• General form of solution:

2

3,

xx

7.2 Systems of Linear Equations in 3 Variables

• Objectives:– Verify the solution of a system of linear

equations in 3 variables– Solve systems of linear equations in 3

variables– Solve problems using systems in 3 variables

What is a system of equations of 3 variables?

• Linear equations that exist in space (not simply in one plane)

• 3 variables: (x,y,z)• 3 lines could intersect at one point, no

points, or infinitely many points (fall on top of each other)

• How do we solve a system of 3 equations? Very much like we do a system of 2 equations! (next slide)

• Use substitution or elimination to solve:• 3x + y – z = -2• x + 2y + z = 4• 2x – y + 2z = -2• ADD the 1st 2 eq: 4x + 3y = 2• Multiply the 1st eq. by (2): 6x + 2y – 2z = -4• NOW add this eq. to 3rd eq: 8x + y = -6• We NOW have 2 eq. with 2 variables:• Solve this new system by multiplying 1st by (-2) &

then adding the 2nd.• Result: (-8x – 6y = -4) + (8x + y = -6):

-5y = -10, y=2• Now solve for x: 8x + 2 = -6, x = -1• Now solve for z: -1 + 4 + z = 4, z = 1• Solution: (-1, 4, 1) (CHECK!!)

7.3 Partial fractions

• Objectives:– Decompose P/Q where Q has only distinct

linear factors– Decompose P/Q, where Q has repeated linear

factors– Decompose P/Q, where Q has a nonrepeated

prime quadratic factor– Decompose P/Q, where Q has a prime,

repeated quadratic factor

What is decomposition of partial fractions?

• Writing a more complex fraction as the sum or difference of simpler fractions.

• Examples:

• Why would you ever want to do this? It’s EXTREMELY helpful in calculus!

22

35353412

43

xxx

x

baba

Distinct linear factors

• Factor the denominator into linear terms

• Each linear term will be the denominator of a separate term (i.e. if there are 3 factors, there will be 3 separate fractions added together)

• Example next slide

Decompose

• Factor denominator: (x+2)(x+1)(x-1)• Now the original expression will be written as the sum of 3

fractions: • A/(x+2) + B/(x+1) + C/(x-1)• To find A,B,C, rewrite this sum as 1 rational expression

(get common denominators)

)1)(1)(2(

232

)1)(1)(2(

)23()2()1(

)1)(2)(1(

)1)(2(

)1)(2)(1(

)1)(2(

)1)(1)(2(

)1)(1(

222

222

xxx

CCxCxBBxBxAAx

xxx

xxCxxBxA

xxx

xxC

xxx

xxB

xxx

xxA

)1)(2(

1527122

2

xx

xx

(example continued)

• Since denominators are equivalent, so must be numerators. Equate like parts.

• Use substitition & addition to solve (add 1st & 3rd equations, then take the result & add to 2nd eq)

• Substitute back in other eq. to solve for all 3 variables

2722

273,27)3(

12,12)( 22

CBA

CBxCBx

CBAxCBAx

3,9,0

9),2(546

)31(273

ACB

CndC

rdstCB

Check your solutions.

• Are these solutions correct?

• At first glance, no, but then with closer examination, we realize the x+1 term in the denominator would cancel with an x+1 term in the numerator, (recall B=0), thus it is correct!

• If we had first simplified the numerator to (x+1)(12x+15) and cancelled the x+1 terms, it may have been easier!

)1)(2(

152712

1

9

2

32

2

xx

xx

xx

What if one on the denominators is a linear term squared?

• This is accounted for by having the nonsquared term as one denominator and having the squared term as another denominator.

• What is one denominator is a linear term cubed? There would be 3 denominators in the decomposition:

32 )()()( term

C

term

B

term

A

What if denominator has a quadratic factor (not reduced to

product of linear factors)?• A quadratic denominator in decomposition would

have a linear numerator:

• To decompose the fraction, you proceed precisely as was done with linear denominators.

123)12)(3(

132522

2

x

CBx

x

A

xx

xx

7.4 Systems of Nonlinear Equations in 2 Variables

• Objectives:– Recognize systems of nonlinear equations in

2 variables– Solve nonlinear systems by substitution– Solve nonlinear systems by addition– Solve problems using systems of nonlinear

equations

Decompose

2

2

2

)3(

6

3

2)4

)3(

9

3

2)3

)3(

3

3

2)2

3

3

3

2)1

xx

xx

xx

xx

2)3(

92

x

x

Non-linear systems: what are they?

• If a linear systems involves finding the intersection of lines, consider what the graph of nonlinear equations might be.

• Graphs could be that of circles, parabolas, ellipses.

• Where might they intersect? A circle and parabola might intersect once, twice, three times, or not at all!

• A line and an ellipse may not intersect, of may have one or 2 points of intersection.

When solving these systems, be aware of ALL possible solutions.

Don’t overlook some!• Graphing the system will provide a clue as

to what the solutions may look like (Be careful when entering some functions. They may need to be entered as piecewise functions.).

Solve:

• Solve using addition (multiply 2nd equations by -9 before adding together)

• Note: this is a circle that sits inside an ellipse & they share the 2 points of intersection.

4,3649 2222 yxyx

)}0,2(),0,2{(:

2,0,05

3699

3649

2

22

22

solution

xyy

yx

yx

7.5 Systems of Inequalities

• Objectives:– Graph a linear inequality in two variables– Graph a nonlinear inequality in two variables– Graph a system of inequalities– Solve applied problems involving systems of

inequalities

Linear inequality in two variables• A linear inequality divides the xy-plane into 2

parts. Either the points on one side of the line make the inequality true or points on the other side do.

• Once the line is determined, select any point on either side to test in the original inequality to determine if that point is a solution or not.

• If the point makes the inequality true, all points on that side are also in the solution set. If the point makes the inequality false, all points on the other side of the line are in the solution set.

System of linear inequalities

• If the 2 lines intersect at one point, the plane is divided into 4 areas. The solution could be found in one of these areas.

• Often graphing and looking for overlapping areas is easier than looking at points in each region.

Nonlinear inequalities

• Consider the graph of a circle. The plan is divided into the area inside the circle, and that outside the circle. Solve it as you would a linear system.

• Consider the graph of other nonlinear inequalities (parabolas, ellipses, hyperbolas). Again, the graph would show that the points that make the inequality true would be found inside OR outside of the graph.

7.6 Linear Programming

• Objectives– Write an objective function describing a

quantity that must be maximized or minimized– Use inequalities to describe limitations in a

situation– Use linear programming to solve problems

Find a linear equation that represents a specific situation, taking into account restrictions

(linear inequalities).

• Each restriction creates a linear inequality

• Plot the equation and the inequalities and examine the vertices (points of intersection of lines)

• The original function is maximized and minimized at one or more corner points.

You are taking a test with a multiple-choice section & a full response section. Each M-C

question is worth 3 pts. & full-response is 7 pts. You have 50 min. to complete the test. M-C

requires 2 min/prob & full-response, 5min/prob. If you answer at most 13 questions

& all your answers are correct, how many of each should you answer for the highest score?

What’s the score?

Example continued• 3x + 7y = score (objective eq.)• 2x+5y < 50, x >0,y>0, x +y < 13• Intersection pt.=(5,8)• Other vertices: (0,10), (0,0),(13,0)

(1st quad)• Put all 4 pairs into objectives eq:• 3(0)+7(10)=70• 3(0)+7(0)=0• 3(13)+7(0)=39• 3(5)+7(8)=71• 5 M-C & 8 full-response for 71 pts.

State the objective equation:Student tickets will be sold for $4 each and adults for $6 each for a play. There are 200 seats in the

auditorium. Every adult will bring at most 3 children. How many of each type of ticket should

be sold to raise the most money?

• 1) Total = 4x + 6y• 2) x + y < 200• 3) y 3x• 4) x>0, y>0