Unit 1 Study Guide Systems of Linear Equations and Inequalities. Part 1: Determine if an ordered pair is a solution to a system

Transcription

1 Unit Stud Guide Sstems of Linear Equations and Inequalities 6- Solving Sstems b Graphing Part : Determine if an ordered pair is a solution to a sstem e: (, ) Eercises: substitute in for and - in for in each equation in the sstem. - - ( ) - ( ) Since the ordered pair onl works in one of the equations, it is not a solution to the sstem.. (,) 5. (, ) 6. (, ) Part : Solving a sstem b graphing e: re-write the equations in slope-intercept form before graphing if necessar.

2 Eercises: Part : Problem Solving e: Mendham Video charges $0 for a membership and $ per movie rental. Chester Video charges $5 for a membership and $ per movie rental. For how man movie rentals will the cost be the same at both video stores? What is that cost? a) Understand the Problem The answer will be the number of videos rented for which the total cost is the same for both video stores. List important information: Mendham Video $0 membership $ per movie Chester Video $5 membership $ per movie b) Make a Plan Write a sstem of equations, one equation to represent the price at each video store. Let be the number of videos rented and be the total cost. Mendham Video 0 Chester Video 5 c) Solve b Graphing The solution to the sstem is where the two lines intersect on the graph. The appear to cross at ( 5, 5 ), so the cost at

3 both video stores will be the same for 5 video rentals and that cost will be $5. d) Look Back Check ( 5, 5 ) using both equations. Mendham Video: 5 ( 5 ) Chester Video: 5 ( 5 ) Eercises:. Mr. Marlone is putting mone in two savings accounts. Account A started with $00 and Account B started with $00. Mr. Marlone deposits $5 in Account A and $0 in Account B each month. In how man months will the accounts have the same balance? What will the balance be?

4 6- Solving Sstems b Substitution Part : Solving a Sstem using Substitution e: Follow the five steps to solving a sstem using substitution. Step : Make sure one or both of the equations is solved in terms of or. The second equation is solved in terms of. Step : Substitute the resulting epression into the other equation. Now ou will have one equation with one variable to solve: ( ) Step : Solve for the variable. Step : Substitute the value of the variable ou just solved for into one of the original equations to solve for the other variable. () 8 5 Step 5: Write the values of the variables ou solved for in steps and as an ordered pair (, ) and check. (, 5 ) Eercises: Equation Equation 5 5 ()

5 Part : Solving a Sstem using Substitution Using the Distributive Propert 5 e: Step : Solve equation for b adding to both sides of the equation. Step : Substitute epression into first equation for. 5 ( ) 5 Step : Solve the equation using distributive propert CAUTION don t forget to distribute the sign with the term 5() 5() Step : Substitute the value of the variable ou just solved for into one of the original equations to solve for the other variable. ( ) 6 5 Step 5: Write the values of the variables ou solved for in steps and as an ordered pair (, ) and check. ( -5, - ) Equation Equation ( ) 5( 5) 6 5 5( ) 5 6 Look in both equations for a variable with a coefficient of or - when deciding which equation to solve for or.

6 Eercises: Solving Sstems b Elimination Part : Solving a Sstem b Elimination Using Addition or Subtraction e: 5 Step : Write the sstem so that like terms are aligned Look for the variables that have the same or opposite coefficients to eliminate, such as - and. Step : Eliminate one of the variables and solve for the other Since the variable we are going to eliminate,, has opposite signs, add the two equations together to eliminate. Step : Substitute the value of the variable into one of the original equations and solve for the other variable. 5( ) Step : Write the answers from Steps and as an ordered pair, (, ), and check ( -, 8 )

7 - 5 - ( 8 ) - 5( - ) ( 8 ) Part : Solving a Sstem b Elimination Using Multiplication First e: Look for a variable in one of the equations that has a coefficient of or -. Step : Write the sstem so that like terms are aligned Step : Multipl one, or both, of the equations b a factor so that the coefficients are the same, or opposites. Eliminate one of the variables and solve for the other. ( ) Step : Substitute the value of the variable into one of the original equations and solve for the other variable. ( ) 6 Step : Write the answers from Steps and as an ordered pair, (, ), and check (, - ) - - ( ) - - ( ) ( - )

8 Eercises: Solving Special Sstems Part : Sstems with No Solution Inconsistent e: Write both equations in Slope-Intercept Form, m b. Compare the slopes and -intercepts. Equation : slope -intercept - Equation : slope -intercept Since the slopes are the same and the -intercepts are different, the lines are parallel. The will never have a point of intersection, so there is No Solution to this sstem. It is Inconsistent. Part : Sstems with Infinitel Man Solution Consistent and Dependent e: 0 Write both equations in Slope-Intercept Form, m b. Compare the slopes and -intercepts. 0 Equation : slope -intercept Equation : slope -intercept

9 Since the slopes and the -intercepts are the same, the lines are also the same. Ever point along one line is a point on the other. There are infinitel man points of intersection, so there are Infinitel Man Solutions to this sstem. It is Consistent and Dependent. Classification of Sstems of Linear Equations Classification Consistent Consistent Inconsistent Independent Dependent # of Solutions Eactl One Infinitel Man None Description Different Slopes Same Slope, Same Slope, Same -int Different -int Graph Two lines, One Line Parallel Lines intersecting at one point Eercises: Solve and classif each sstem Solving Linear Inequalities Part : Determine if an ordered pair is a solution to the inequalit e: ( 7,; ) 7 substitute 7 in for and in for in the inequalit. 6 Since the ordered pair makes the inequalit a true statement, it is a solution to the inequalit.

10 Part : Graphing an inequalit in the coordinate plane e: < Step : Solve the inequalit for (slope-intercept form) Step : Graph the line as ou would an linear equation, start with the - intercept and use slope to find more points. Use a solid line for all inclusive signs, or. Use a dotted/dashed line for all non-inclusive signs, < or >. Step : Shade the half-plane above the line for half-plane below the line for < or. > or. Shade the Step : Pick a point within the shaded region, a boundar point, to check. < - dotted/dashed line - shade below Part : Writing an inequalit from a graph e:

11 Step : Identif the slope and -intercept from the graph slope - intercept Step : Determine inequalit sign from the tpe of line graphed: - dotted/dashed is > or < ( non inclusive ) - solid is or ( inclusive ) the line is solid so it is an inclusive smbol Step : Determine direction of inequalit sign b shading: - shaded below <or - shaded above >or the shading is below, so the smbol is Step : Write the inequalit using all of this information Eercises:. Is the ordered pair a solution?. Graph: > (,; ). Write an inequalit for the graph below:

12 6-6 Solving Sstems of Linear Inequalities Part : Determine if an ordered pair is a solution to the sstem < e: (,) Eercises: < < - ( ) < substitute in for and in for in each inequalit in the sstem. Since the ordered pair works in both of the inequalities, it is a solution to the sstem.. ( 0,) <. ( 0,0) > > Part : Solving a sstem b graphing 8 e: > 8 > > Step : Re-write each inequalit in slope-intercept form, if necessar Step : Graph each inequalit separatel, don t forget about solid or dotted lines and our shading. Step : Highlight the area where shading from both inequalities overlap. This is our solution.

13 Eercises:. >. > 7 6