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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: • Replace the ordered pair in the system for x and y. • Is the equation true? • If so, then it is a solution. • If not, then it is not. • Note: It must be true for both equations.
Solving a System of Linear Equation Ways to solve: • By graphing- The point where the lines intersect is the solution. • By Substitution
Solving by Substitution • Solve by substitution: 5x – 4y = 9 X – 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: • By graphing- The point where the lines intersect is the solution. • By Substitution • By Elimination
Ex. 2: Solving a System by Addition • 3x + 2y = 48 • 9x – 8y = -24 • Steps: • Rewrite both equations in the form of AX + BY = C. • If necessary, multiply either equation or both equations by appropriate numbers so that the sum of the x-coefficients or y-coefficients = 0. • Add the equations • Solve for one variable. • 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 + 600 Income is price times copies sold. y = 1.5x To 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: • Reduce the system to 2 equations in 2 variables. • Solve the resulting system of 2 equations in 2 variables. • Use back-substitution in 2 variables to find the value of the second variable. • 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: • Solve one equation for one variable in terms of the other. • Substitute the expression from Step 1 into the other equation. • Solve the resulting equation containing one variable. • Back substitute the obtained values into the equation. • 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= 13 X2 + y2 = 10
Guided Practice: • Solve the system: Y = X2 + 5 X2 + Y2 = 25
Example 4: • Solve the system: Y = X2 + 3 X2 + Y2 = 9
Guided Practice: 3x2 + 2y2 = 35 4x2 + 3y2 = 48
Summary: Answer in complete sentences. • Solve the following systems by the method of your choice. Then explain why you chose that method. • 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: • Replace Inequality with = sign and graph the linear equation. • Choose a test point from one of the half planes and not from the line. Substitutes its coordinates into the inequality. • Then shade the plane that meets the conditions.
Practice: • Graph 4x – 2y > 8
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: • Replace inequality with = sign and graph. • Choose a test point from one of the regions not on the circle. • 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: • Replace each inequality symbol with an equal sign. Graph using the x and y-intercepts of each equation. • 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.
Practice: • Graph the solution set of the system below.
Example 6: Graphing System of Inequalities • Graph the solution set of the equation: • Steps: • Graph the first inequality. Note it’s a solid parabola. • Then graph the second inequality. • Note the points of intersection of the inequalities. • Now test points and shade the region that makes the inequalities true.
Practice: • Graph the system of inequalities.
Example 7: Graphing a System of Inequalities • Graph the solution set of the system:
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?