Systems of Equations and Inequalities

Chapter 7

Aim #7.1: How do we solve systems of linear equations?

• All equations in the form of Ax+ By = C form a straight line when graphed.

• Two such equations are or a linear system. systems of equations

• A solution to a system of a linear equations in two variables is an ordered pair that satisfies both equations.

Example 1:

• Determine if each ordered pair is a solution of the following system.

a. (4, -1) b. (-4, 3)• X + 2Y = 2• X - 2Y = 6

• Steps:1. Replace the ordered pair

in the system for x and y. 2. Is the equation true?3. If so, then it is a solution.4. If not, then it is not.5. Note: It must be true for

both equations.

Solving a System of Linear Equation

Ways to solve:1.By graphing- The point where the lines

intersect is the solution.2.By Substitution

Solving by Substitution• Solve by substitution:

5x – 4y = 9X – 2y = -3

• Steps:1.Solve either of the equations

for one variable in terms of the other.

2. Substitute the expression from step 1 into the other equation.

3. Solve the resulting equation.4. Then substitute the vale into

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

Try:

• Solve by the substitution method:• 3X + 2Y = 4• 2X + Y = 1

Solving a System of Linear Equation

Ways to solve:1.By graphing- The point where the lines

intersect is the solution.2.By Substitution3.By Elimination

Ex. 2: Solving a System by Addition

• 3x + 2y = 48• 9x – 8y = -24

• Steps:1. Rewrite both equations in the

form of AX + BY = C.2. If necessary, multiply either

equation or both equations by appropriate numbers so that the sum of the x-coefficients or y-coefficients = 0.

3. Add the equations4. Solve for one variable.5. Then substitute back into one of

the original equations and solve for the other variable.

Guided Practice:

• Solve by the elimination method:• 2x = 7y – 17• 5y = 17 – 3x

Analyzing Special Types of Systems

• When lines are parallel there are no points of intersection. So the system of linear equations has no solution.

• When the equations of the lines are the same then you have infinitely many

Example 3: A System with No Solution

• Solve the system:• 4X + 6Y = 12• 6X + 9Y = 12

Example 4: Infinitely Many Solutions

• Solve the System.• Y = 3X – 2

• 15X – 5Y = 10

Applications:• Example 1: A metalworker has some ingots of

metal alloy that are 20% copper and others that are 60% copper. How many kilograms of each type of ingot should the metalworker combine to create 80 kg of a 52% copper alloy?

• Let g = mass of the 20% alloy• m = mass of the 60% alloy• Mass of the alloys: g + m = 80• Mass of copper: 0.2g + 0.6m= .52(80)

• Now solve for g and m.

Break- Even Problems• Suppose a model airplane club publishes a newsletter.

Expenses are $.90 for printing and mailing each copy, plus $600 total for research writing. The price of the newsletter is $1.50 per copy. How many copies of the newsletter must the club sell to break even?

• Let x = the number of copies• y = the amount of dollars of expenses or income Expenses are printing costs plus research and writing. y = 0.9x + 600Income is price times copies sold. y = 1.5xTo find out how many copies you need to sell solve for x.

Summary: Answer in complete sentences.

• 3- What are three ways to solve systems of equations?

• 2- Identify 2 elimination strategies.• 1-Solve:

Suppose an antique car club publishes a newsletter. Expenses are $.35 for printing and mailing each copy, plus $770 total for research and writing. The price of the newsletter is $.55 per copy. How many copies of the newsletter must the club sell to break even?

Aim #7.2 How do we solve systems with three variables?

• An equation in the form of Ax + By +Cz = D, is linear equation with 3 variables.

• Linear variables are: x, y and z are the variables.

• Example: x + 2y – 3z = 9

Example 1:

• Show that the ordered triple (-1, 2, -2) is a solution of the system:

X + 2y – 3z = 9 2x – y + 2z = -8

- x + 3y – 4z = 15

Try:

• Show that the ordered triple (-1,-4 , 5) is a solution of the system:

X - 2y + 3z = 22 2x – 3y - z = 5

3x + y – 5z = -32

Example 2: Solving a System in Three Variables

• Solve the system:• 5x – 2y – 4z = 3• 3x + 3y + 2z = -3• -2x + 5y + 3z = 3

Steps:1. There are many ways to approach. The central idea is to take

two equations and eliminate the same variable from both pairs.2. Solve the resulting system of two equations in 2 variables.3. Use back-substitution to find the value of the second variable.4. Solve for the third –variable.

Guided Practice:

• Solve the system:• X + 4Y – Z = 20• 3X + 2Y + Z = 8• 2X – 3Y + 2Z = -16

Example 3: Solving a System w/a Missing Term

• Solve the system:• X + z = 8• X + y + 2z = 17• X + 2y + z = 16

• Steps:1. Reduce the system to 2

equations in 2 variables.2. Solve the resulting system of 2

equations in 2 variables.3. Use back-substitution in 2

variables to find the value of the second variable.

4. Then find the third variable.

Practice:

• Solve the system:• 2y – z = 7• X + 2y + z = 17• 2x - 3y + 2z = -1

Summary: Answer in complete sentences.

• What and how do you solve a system of linear equations with 3 variables?

• Give an example from your class work to support your explanation.

• Determine if the following statement makes sense, and explain your reasoning.A system of linear equations in 3 variables, x, y, and

z cannot contain an equation in the form y = mx + b .

Aim #7.4 How do we solve systems of nonlinear equations in 2 variables?

• A system of two nonlinear equations in two variables, also called a nonlinear system contains at least one equation that cannot be expressed in the form Ax + By = C.

• Example:• X2 = 2y + 10• 3x – y = 9

• A solution of a nonlinear system in two variables is an ordered pair of real numbers that satisfies both equations in the system.

• The solution set of the system is the set of all such ordered pairs.

• Unlike linear systems, the graphs can be circles, parabolas or anything other than two lines.

• To solve nonlinear systems we will use the substitution method and the addition method.

Example 1: Solving a Nonlinear System by the Substitution Method

• Solve:• X2 = 2Y + 10• 3x – Y = 9

Steps 1:1. Solve one equation for one variable in terms of the other.2. Substitute the expression from Step 1 into the other

equation.3. Solve the resulting equation containing one variable.4. Back substitute the obtained values into the equation.5. Check the proposed solution.

Guided Practice:

• Solve by the substitution method:• X2 = y -1• 4x – y = -1

Example 2:

• Solve by the substitution method:• X – Y = 3

• (X – 2)2 + (Y + 3)2 = 4 • (Note: This is a circle, with the center at (2, -3)

and radius 2.)

Practice:

• Solve by the substitution method:X + 2Y = 0

(X – 1)2 + (Y - 1)2 = 5

Example 3: Solving a Nonlinear System using the Addition Method• Solve the system:

4x2 + y2= 13X2 + y2 = 10

Guided Practice:

• Solve the system:Y = X2 + 5

X2 + Y2 = 25

Example 4:

• Solve the system:Y = X2 + 3X2 + Y2 = 9

Guided Practice:

3x2 + 2y2 = 354x2 + 3y2 = 48

Summary: Answer in complete sentences.• Solve the following systems by the method of

your choice. Then explain why you chose that method.

a. X – 3y = -5 X2 + Y2 - 25 = 0

b. 4X2 + XY = 30 X2 + 3XY = -9

Aim #7.5: How do we solve system of inequalities?

• Graphing a linear Inequality in Two Variables• Graph 2x – 3y > 6

• Steps:1. Replace Inequality with = sign and graph the linear

equation.2. Choose a test point from one of the half planes and

not from the line. Substitutes its coordinates into the inequality.

3. Then shade the plane that meets the conditions.

Practice:

• Graph 4x – 2y > 8

Example 2: Graph the inequality in Two Variables

• Graph

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y > −2

3x

Example 3:

• Graph the following inequalities.• Hint: Graph y = 3 and x = 2. Graph the region

that meet the conditions below.a. Y< - 3 (The values less than and equal to 3)

b. X > 2( The values greater than 2)

Example 4: A Nonlinear Inequality

• Graph x2 + y2 < 9

Steps: 1.Replace inequality with = sign and graph.2.Choose a test point from one of the regions

not on the circle. 3.Shade the region that meets the conditions.

Practice:

• Graph x2 + y2 >16

Example 5:Graphing Systems of Linear Inequalities

• Graph the solution set of the system.

• Steps:1. Replace each inequality symbol with an equal sign.

Graph using the x and y-intercepts of each equation.2. Note the first equation the line should be ..? Whereas the second equation’s line should be…?3. Test points in each region to see which section is a

solution to both inequalities.

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x − y <1

2x + 3y ≥12

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Practice:

• Graph the solution set of the system below.

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x − 3y < 6

2x + 3y ≥ −6

⎧ ⎨ ⎩

Example 6: Graphing System of Inequalities

• Graph the solution set of the equation:

• Steps:1. Graph the first inequality. Note it’s a solid

parabola.2. Then graph the second inequality.3. Note the points of intersection of the inequalities.4. Now test points and shade the region that makes

the inequalities true.

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y ≥ x 2 − 4

x − y ≥ 2

⎧ ⎨ ⎩

Practice:

• Graph the system of inequalities.

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y ≥ x 2 − 4

x + y ≤ 2

⎧ ⎨ ⎩

Example 7: Graphing a System of Inequalities

• Graph the solution set of the system:

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x − y < 2

−2 ≤ x < 4

y < 3

⎧

⎨ ⎪

⎩ ⎪

Summary:Answer in complete sentences.

• What is a linear inequality in two variables? Provide an example with your description.• How do you determine if an ordered pair is a

solution of an inequality in two variables x and y?

• What is the difference between a dash and a solid line in graphing an inequality?

• What is a system of linear inequalities?