Solution of the System of Linear Equations: any ordered pair in a system that makes all equations true.

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1 Definitions: Sstem of Linear Equations: or more linear equations Sstem of Linear Inequalities: or more linear inequalities Solution of the Sstem of Linear Equations: an ordered pair in a sstem that makes all equations true. Solution of the Sstem of Linear Inequalities: an ordered pair in a sstem that makes all inequalities true. Ke Concepts: One Solution: Different Slopes No Solution: Lines are parallel, same slopes but different -intercepts Infinitel Man Solutions: Lines are the same, same slope and same -intercepts Solving Sstem of Equations b Graphing In order to solve a sstem of equations b graphing Slope-Intercept Form ( = mx + b): o Graph each equation using the -intercept (b) and the slope (m). Standard Form (Ax + B = C) o Find the slope (-A/B) o Find the x-intercepts & -intercepts o Graph using the slope, x-intercepts & -intercepts In order to ensure that both lines are long and will connect, make sure to plot the slope of each line several times in both directions. The point where lines intersect is the solution the sstem. Check to see if the solution makes both equations true. Examples: Solve Sstem of Equations b Graphing Solve b graphing. Check our solution.. = x + = x + Slope Intercept Form: Step : Find slope & -intercepts b = (0, ) m = ( ) b = (0, ) m = ( ) Step : Determine # of solutions Since slopes and -intercepts are different, there is solution. Step : Graph equations & look for point of intersection x Step : Check solution (, ) = x + = x + = ()+ = () + = = + = Rev A

2 . x + = x + = 0 Standard Form: Step : Find slope, x-intercepts & -intercepts b = (0, ) m = ( ) b = (0, 0) m = ( ) Step : Determine # of solutions Since slopes and -intercepts are different, there is solution. Practice: Solve Sstem of Equations b Graphing Solve b graphing. Check our solution.. = x + = x Solution: Step : Graph equations & look for point of intersection x Step : Check solution (-, ) -x + = x + = 0 -(-)+ = (-) + = 0 + = - + = 0. x + = 8 x + = Solution: x x Example: Word Problem. Suppose ou have $0 in our bank account. You start saving $ each week. Your friend has $ in his account and is saving $0 each week. Assume that neither ou nor our friend makes an withdrawals. a. After how man weeks will ou and our friend have the same amount of mone in our accounts? b. How much mone will each of ou have? Rev A

3 Solving Sstem of Inequalities b Graphing In order to solve a sstem of inequalities b graphing Repeat the same steps for graphing equations Use the inequalit smbol to draw either dotted or solid lines and shade the appropriate area. The area that is shaded b both inequalities is the solution to the sstem of inequalities. Check to see if the solution makes both inequalities true. Example: Solve Sstem of Inequalities b Graphing Solve b graphing. Check our solution. 6. > x x + < Step : Find relevant information to graph Ineq : Slope Intercept Form: Find slope & -intercepts b = (0, -) m = ( ) Step : Graph inequalities & look for overlapping region Step : Find a solution & check it. Ineq : Standard Form: Find slope, x-intercepts & -intercepts b = (0,) m = - ( ), x-int = (,0) x Practice: Solve Sstem of Inequalities b Graphing 7. < x + x - 8. > x -x + x x 9. -x + x x + - x + x x Rev A

4 Evaluating Expressions When asked to evaluate an expression, ou simpl replace the variable with the value and follow our order of operations (PEMDAS) Examples: Evaluating Expressions Evaluate the following expressions for the given values.. Y = -x + for x = Step : replace x with Y = -() +. Y = x 6 for x = Step : Follow PEMDAS and simplif Y = - + = -. Y = -x + for =. Y = x - 6 for =. x + = 0 for x = 6. x + = 0 for = Solving for a variable When asked to solve an equation for an indicated variable, Follow same steps as solving an equation. Note: result will tpicall be an algebraic expression Examples: Solving for indicated variables Solve each equation for the indicated variables 7. Y = -x + solve for x x 8. d = + b solve for x a x = solve for 0. ax + bx - = 0 solve for x. x + 6 = 0 solve for x. x + = 0 solve for x Rev A

5 Substitution Method Replace variable with an equivalent expression containing the other variable Solve for other variable using process for solving a variable equation Substitute value of solved variable to find the value of the other variable. Check solution Example: Solve b Substitution. = -x + 8 = x + 7 Step : Write equation containing variable & solve. = -x + 8 x + 7 = -x + 8 Substitute x + 7 for +x +x Add x to both sides x + 7 = Subtract 7 from both sides x = Divide both sides b x = 0. Step : Solve for other variable in either equation. Step : Check solution (0., 7.) = x = -(0.) = 7. = (Substitute 0. for x) 7. = = = 7. = x = -6x + = -7 Step : Solve variable in terms of the other variable. -6x + = -7 +6x +6x Add 6x to both sides = 6x -7 Step : Write equation containing variable & solve. (6x 7) + x = Substitute 6x 7 for in + x =. 8x + x = Use distributive propert. 0x = Combine like terms + + Add to both sides 0x = 0 0 Divide both sides b 0 x =. Step : Solve for other variable in either equation. Step : Check solution (., 0.) -6x + 7 = 7 + x = -6x + = -7-6(.) + = -7 Substitute. for x (0.) + (.) = -6(.) + 0. = = = = Add 7. to both sides. = -7 = -7 = 0. Rev A

6 Practice: Solve b Substitution. = x 7x = 6. = x + = -x + 7. x + = - x + = 8. = x 6x = 8 9. = x + x = + 0. x = x = 0 Example/Practice: Real-World Problems. School committee is planning after-school trip b 9 people to competition at another school. There are 8 drivers available and tpes of vehicles (school buses & minivans). School buses seat people & minivans seat 8 people each. How man buses & minivans will be needed? Step : Identif variables. Let b = school buses and m = minivans Step : Write equations. Drivers b + m = 8 People b + 8m = 9 Step : Solve algebraicall. Step : Check solution. A shop sells gift wrap for $ per package and greeting cards for $0 per package. If the shop sells 0 packages in all and receives a total of $08, how man packages of gift wrap and greeting cards were sold? 6 Rev A

7 Elimination Method Add or subtract equations to eliminate a variable. o If not possible, then multipl or both equations b number(s) that will allow ou to add/subtract the equations. Solve for the nd variable. Plug in value of the nd variable into one of the equations to solve for the st variable. Check solution Example: Solve b Elimination (Add/Subtract Equations). Solve b elimination: x 6 = - x + 6 = 8 Step : Eliminate because the sum of the coefficients of = 0 x 6 = - + x + 6 = 8 8x + 0 = 6 (addition propert of equalit) x = (Solve for x) Step : Check Solution (, 7) () 6(7) = - () + 6(7) = 8 0 = = 8 - = - 8 = 8 Step : Solve for eliminated variable using an of equations. x + 6 = 8 () + 6 = 8 (Substitute for x) = 8 (Simplif and then solve for ) = 7 Practice: Solve b Elimination (Add/Subtract Equations). 6x = -6x + =. x + = -x + 9 = 6. x + = 7 6x = -9 Example: Multipling Equation: 7. Solve b elimination: x + = - 0x + = Step : Eliminate variable Start with given ss To prepare for eliminating x, multipl st eq b Subtract equations to eliminate x. x + = - (x + = -) 0x + = -0 0x + = 0x + = 0x + = Step : Solve for = - = = - Step : Solve for eliminated variable using either equation: x + = - x + (-6) = - (Substitute 6 for ) x 0 = - x = 8 x = (Solve for x) Step : Check solution (, -6) () + (-6) = - 0() + (-6) = = = - = - = 7 Rev A

8 Practice: Multipling Equation: 8. x + = - 7x = 7 9. x + 6 = -6 -x = x + 8 = 0 8x + = Example/Practice: Real-World Problems. EAHS sells a total of 9 tickets for a basketball game. An adult ticket is $. A student ticket costs $. $70 is collected in ticket sales. Write and solve a sstem to find the number of each tpe of ticket sold. Step : Define the variables Let a = # of adult tickets s = # of student tickets Step : Write equations ( for # of tickets & for value of tickets) a + s = 9 (total # of tickets) a + s = 70 (total amount of sales) Step : Solve algebraicall. Step : Check solution. Tickets for a softball game at EAHS cost $ for adults and $ for students. The attendance that da was 9, and $067 was collected. Write and solve sstem of equations to find the number adults and number of students that attended the game. 8 Rev A

9 Real-World Applications When dealing with word problems, there will tpicall be tpes of equations written When given a total # and a value ou will get Quantit Equation (i.e. the total number of items ou have) Qualit Equation (i.e. what is each item worth) Depending on the equations, the sstem can be solved in an of the following methods. Pick which one makes sense for the given problem Solve b Graphing Solve b Substitution Solve b Elimination Example:. You have 0 coins, all dimes and nickels. The value of the coins is $.7. How man dimes and quarters do ou have? Let d = dimes and q = quarters Equation : Quantit Equation d + q = 0 Equation : Qualit Equation Dimes are worth.0 and quarters are worth..0d +.q =.7 Looking at the equations, I can easil use substitution. Step : re-write equation Step : replace q with 0-d in equation Q = 0 d.(0 d) +.0d =.7 Step : Solve for d Step : Solve for q. -.d +.0d =.7 q + d = 0 -.d = -.7 q + = 0 d = q = Step : Check solution (,) + = 0.() +.0() =.7 0 = =.7.7 =.7. An amusement park charges admission plus a fee for each ride. Admission plus rides costs $0. Admission plus rides cost $6. What is the charge for admission? For each ride? Let a = admission, r = rides Equation : a + r = 0 Equation : a + r = 6 Looking at the equations, I can easil use elimination a + r = 0 - a + r = 6 -r = -6 r = Now solve for a: a + () = 0 => a + = 0 => a = 6 Check solution (6,) 6 + () = () = 6 9 Rev A

10 Practice:. You have a total of coins, all nickels and quarters. The total value is $.8. Write and solve a sstem of equations to find the number of nickels (n) and number of quarters (q) that ou have. 6. A local pizzeria sells a small pizza with topping for $6.00 and a small pizza with toppings for $8.00. What is the charge for a plain small pizza? What is the charge for each topping? 7. A new car dealerships sells cars and trucks in a ratio of 7 to. Last month the dealership sold 8 cars and trucks. How man cars and how man trucks were sold? 8. Suppose ou bought supplies for a part. Three rolls of streamers and part hats cost $0. Later, ou bought rolls of streamers and part hats for $. How much did each roll of streamers cost? How much did each part hat cost? 9. The sum of numbers is 0. Their difference is. What are the numbers? 0 Rev A