Reteaching Solving Equations To solve an equation that contains a variable, find all of the values of the variable that make the equation true. Use the equalit properties of real numbers and inverse operations to rewrite the equation until the variable is alone on one side of the equation. Whatever remains on the other side of the equation is the solution. Addition Propert of Equalit To isolate w on one side of the equation, add 7 to each side. w 7 5 7 7 w 5 8 Multiplication Propert of Equalit To isolate w on one side of the equation, multipl each side b. Subtraction Propert of Equalit To isolate 5z on one side of the equation, subtract from each side. 5z 5 5z 5 0 Division Propert of Equalit To isolate z on one side of the equation, divide each side b 5. w 5 8 5z 0 5 5 5 z 5 w 5 6 Eercises Solve each equation.. + = 8. p - 9 =. - + r = 0. 8 = 5 + k 7 5. 9q = 7 6. 6 t = - 7. 5 = - d 7 5 8. 9 = 0m.9 9. g - = 9 0. n 8 + 9 = 8. 7 = -5 - c 8. = f + 9.5 Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
Reteaching (continued) Solving Equations To solve an equation for one of its variables, rewrite the equation as an equivalent equation with the specified variable on one side of the equation b itself and an epression not containing that variable on the other side. The equation a - b = + b defines a relationship between a, b, and. What is in terms of a and b? Use the properties of equalit and the properties of real numbers to rewrite the equation as a sequence of equivalent equations. a - b = + b a - b = ( + b) Multipl each side b. a - b = ( + b) Simplif. a - b = + b Distributive Propert a - = b + b Add and subtract to get terms with on one side and terms without on the other side. a - = 5b Simplif. (a - ) = 5b Distributive Propert = 5b a - Divide each side b a -. The final form of the equation has on the left side b itself and an epression not containing on the right side. Eercises Solve each equation for the indicated variable.. m - n = m + n, for m m = n. (u + v) = w - 5u, for u 5. a + b = c + d, for = d a b c 6. k ( + z) = ( - 5), for u = w 6v 7 kz + 0 = k 7. 9. r + s =, for r r = 6s 8. f + 5 g = - f g, for f + k j = j k a -, for = 0. b + = a +, for f = = 5g 8 + g a + b ab b + Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
Reteaching Solving Inequalities As with an equation, the solutions of an inequalit are numbers that make it true. The procedure for solving a linear inequalit is much like the one for solving linear equations. To isolate the variable on one side of the inequalit, perform the same algebraic operation on each side of the inequalit smbol. The Addition and Subtraction Properties of Inequalit state that adding or subtracting the same number from both sides of the inequalit does not change the inequalit. If a 6 b, then a + c 6 b + c. If a 6 b, then a - c 6 b - c. The Multiplication and Division Properties of Inequalit state that multipling or dividing both sides of the inequalit b the same positive number does not change the inequalit. If a 6 b and c 7 0, then ac 6 bc. If a 6 b and c 7 0, then a c 6 b c. What is the solution of ( + ) - 5 -? Graph the solution. Justif each line in the solution b naming one of the properties of inequalities. + 6-5 - Distributive Propert + - Simplif. + Addition Propert of Inequalit 0 Subtraction Propert of Inequalit 5 Division Propert of Inequalit To graph the solution, locate the boundar point. Plot a point at = 5. Because the inequalit is less than or equal to, the boundar point is part of the solution set. Therefore, use a closed dot to graph the boundar point. Shade the number line to the left of the boundar point because the inequalit is less than. Graph the solution on a number line. 5 6 7 8 Eercises Solve each inequalit. Graph the solution.. + ( - ) 7 +. - ( - ) Ú 5 - ( + ) # 0 5 0 Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
Reteaching (continued) Solving Inequalities The procedure for solving an inequalit is similar to the procedure for solving an equation but with one important eception. The Multiplication and Division Properties of Equalit also state that, when ou multipl or divide each side of an inequalit b a negative number, ou must reverse the inequalit smbol. If a 6 b and c 7 0, then ac 6 bc. If a 6 b and c 7 0, then a c 6 b c. What is the solution of - ( - ) 6 + 5? Graph the solution. Justif each line in the solution b naming one of the properties of inequalities. - ( - ) 6 + 5 - + 6 + 5 Distributive Propert - + 6 + 5 Simplif. - 6 Subtraction Propert of Inequalit 7 - Division Propert of Inequalit The direction of the inequalit changed in the last step because we divided both sides of the inequalit b a negative number. Graph the solution on a number line. 0 Eercises Solve each inequalit.. - -(- - ) #. 7-7( - 7) 7 - + 5 * 5 5. 7( + ) - Ú + ( + ) " 6 6. - 6 6-5 + Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
Reteaching Linear Functions and Slope-Intercept Form You can use the slope-intercept form to write equations of lines. The slope-intercept formula is = m + b, where m represents the slope of the line, and b represents its -intercept. The -intercept is the point at which the line crosses the -ais. The slope of a horizontal line is alwas zero, and the slope of a vertical line is alwas undefined. What is the equation of the line that contains the point (, -) and has a slope of -? - = ( - )() + b To find b, substitute the values - for m, for, and - for, into the slope-intercept formula. - = - + b Multipl. = b Add to each side and simplif. = - + Substitute - for m and for b into the slope-intercept formula. Eercises Write an equation for each line.. m = ; contains (, ). m = -; contains (, 7). m = 0; contains (, 0) 5 0 5 5 5 0. m = -; contains (-5, -) 5. m = ; contains (-, -) 6. m = 0; contains (0, -7) 5 7 5 5 7 7. m = 8; contains (5, 0) 8. m = -; contains (0, 7) 9. m = 0; contains (, 8) 5 8 0 5 7 5 8 0. m = ; contains (, 5). m = 7; contains (, ). m = -; contains (, -6) 5 5 7 9 5. 5 O. O 5. O 5 5 = + Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
Reteaching (continued) Linear Functions and Slope-Intercept Form You can graph a linear equation if ou know the slope and the -intercept. Write the linear equation in slope-intercept form. Plot the -intercept. Plot a second point using the slope. Draw a line through the two points. What is the graph of + = 8? Write the linear equation in slope-intercept form. + = 8 = - + 8 = - + Original equation Slope-intercept form The slope is - and the -intercept is (0, ). Plot the -intercept. Use the slope to plot a second point. Then draw a line through the two points. Eercises Graph each equation. 6 (0, ) O 6. - + = 6 7. + = 8. - = 6 O 6 O 6 O 9. - + - = 0 0. + 7 = -. - 6 = 0 6 O 6 O 6 6 O Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
Reteaching Families of Functions Horizontal and Vertical Translations If h and k are positive numbers, then g() = f () + k shifts the graph of f () up k units. g() = f () k shifts the graph of f () down k units. g() = f ( + h) shifts the graph of f () left h units. g() = f ( + h) shifts the graph of f () right h units. 0 0 How can ou represent each translation of = graphicall?. a. g () = - Shift the graph of f () = down units. b. h() = + Shift the graph of f () = left unit. f() h() O g() f() g(). a. g() = - + Shift the graph of f () = right units and up unit. b. h() = + - Shift the graph of f () = left units and down units. O h() Eercises Identif the tpe of translation of f () =.. g() = 0-0 right units. g() = 0 0 +. g() = 0 0 - down units. g() = 0 + 0 up unit left units O Graph each translation of f () =. 5. g() = 0-0 - 5 6. g() = 0 + 0 + 6 O Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
Reteaching (continued) Families of Functions Reflection, Stretching, and Compression If h and k are positive numbers, then g () = f () reflects the graph of f () in the ais. g () = f ( ) reflects the graph of f () in the ais. g () = af (), a +, is a vertical stretch of the graph of f (). g () = af (), 0 * a *, is a vertical compression the graph of f (). What transformations change the graph of f () to g()? f () = 5 g() = -(5) There are two transformations. First transformation: The graph of = (5) is the graph of f () = 5 stretched verticall b a factor of because a = and a 7. Second transformation: The graph of g() = -(5) is the graph of = f (5) reflected in the -ais because the sign of g() has changed. So, the graph of g() is the graph of f () stretched verticall b a factor of and reflected over the -ais. g() (5) (5) f() 5 O Eercises Describe the transformations of f () that produce g(). 7. f () = -5 verticall compressed 8. f() = b a factor of g() = 5 and g() = reflected in the -ais + verticall compressed b a factor of and translated up units Graph f () and g() on the same coordinate plane. 9. f () = 0. f () = g() = -( - ) g() = ( - ) g() ( ) f() f() 8 8 O 8 g() ( ) Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
Reteaching Absolute Value Functions and Graphs A function of the form = a 0 - h 0 + k is an absolute value function. The graph of = a 0 - h 0 + k is an angle; its verte is located at the point (h, k). What is the graph of = 0 + 0 -? This function is in the general form = a 0 - h 0 + k where a =, h = -, and k = -. The verte is (h, k) or (-, -). Now make a table showing several points on the graph. Choose values of on both sides of the verte. 5 Plot the verte and the points from the table. Connect the points to graph the function. Eercises O Make a table of values for each equation. Then graph the equation.. = 0 0. = 0-0 Table of values ma var. 6 Table of values ma var. 6 O. = 0 + 0. = 0-0 + Table of values ma var. O Table of values ma var. O O 5. = 0 + 0-5 6. = - 0-0 + Table of values ma var. O 6 Table of values ma var. 6 O Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
Reteaching (continued) Absolute Value Functions and Graphs The table below shows the transformations of the graph of the parent function = 0 0. A translation is a transformation that shifts a graph verticall or horizontall without changing the graph s shape, size, or orientation. Vertical Translation of 5 5 k If k. 0, translate UP k units. If k, 0, translate DOWN k units. Vertical Stretch and Compression of 5 5 a, a. 0 If a., the graph is narrower. This is a vertical stretch of the parent graph. If a,, the graph is wider. This is a vertical compression of the parent graph. Horizontal Translation of 5 5 h If h. 0, translate to the RIGHT h units. If h, 0, translate to the LEFT h units. Reflection of 5 5 The graph of the parent function is reflected in the -ais. Compare = - 0 + 0 + with the parent function f () = 0 0. Without graphing, find the verte, ais of smmetr, and transformations. First find the verte: This function is in the general form = a 0 - h 0 + k where a = -, h = -, and k=. The verte is (-, ). The ais of smmetr is = -. Describe the transformation of the parent function f () = 0 0 : k =, so the graph is translated up units. a = -, so the graph is reflected in the -ais AND verticall stretched b a factor of. h = -, so the graph is translated to the left unit. Eercises Without graphing, identif the verte, ais of smmetr, and transformations from the parent function f () =. 7. = 0-0 + 8. = 0 0 9. = 0 + 0 - (, ); = ; translated unit right and units up 0. = - 0 0. = 0 0 +. = 0-5 0 + (0, 0); = 0; reflected in the -ais, verticall compressed b a factor of (0, 0); = 0; vertical stretch b a factor of (0, ); = 0; verticall stretched b a factor of, translated units up (, ); = ; translated unit left, verticall stretched b a factor of, units down (5, ); = 5; translated 5 units right, verticall stretched b a factor of, units up Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
Reteaching Two-Variable Inequalities A linear inequalit in two variables is an inequalit whose graph is a region of the coordinate plane bounded b a line. This line is the boundar. If the boundar is included in the solution of the inequalit, it is drawn as a solid line. If the boundar is not part of the solution of the inequalit, it is drawn as a dashed line. What is the graph of 6 -? 6 - Ú - 6 O 8 To graph the boundar line, write the inequalit in slope-intercept form as if it were an equation. The boundar line is solid if the inequalit contains or Ú. The boundar line is dashed if the inequalit contains 6 or 7. Graph the boundar line = - 6 as a solid line. 0 Ú (0) - 6 Since the boundar line does not contain the origin, substitute the point (0, 0) into the inequalit. 0 Ú -6 Simplif. The resulting inequalit is true. Shade the region that contains the origin. If the resulting inequalit were false, then ou would shade the region that does not contain the origin. O 8 Eercises Graph each inequalit.. 7. + 6. 6 + O. - 5 + 5. 6-6. Ú 5 O Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
Reteaching (continued) Two-Variable Inequalities To graph two-variable absolute value inequalities, graph the boundar line. Then pick a test point and shade appropriatel. What is the graph of - 6 0 + 0? - 6 0 + 0 To graph the boundar line, write the inequalit in terms of as if it were an equation. The boundar line is dashed because the 7-0 + 0 + inequalit contains 7. O Graph the boundar line = - 0 + 0 +. 0 Ú - 0 0 + 0 + Now pick a test point. Because the boundar line does not contain the origin, substitute the point (0, 0) into the inequalit. 0 Ú Simplif. The resulting inequalit is untrue. O Shade the region that does not contain the origin. If the resulting inequalit were true, then ou would shade the region that does contain the origin. Eercises Graph each absolute value inequalit. 7. 0 0 8. 7 0-0 9. Ú - 0 0 7 7 5 5 O O O 6 0. 6 0 + 0 -. 0 0-5 O 6 O O 6. - 7 0-6 0 6 Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
Reteaching Solving Sstems Algebraicall Follow these steps when solving b substitution. Step Solve one equation for one of the variables. Step Substitute the epression for this first variable into the other equation. Solve for the second variable. Step Substitute the second variable s value into either equation. Solve for the first variable. Step Check the solution in the other original equation. + = 0 What is the solution of the sstem of equations? e + = 0 Step = - + 0 Solve one equation for. Step (- + 0) + = 0 Substitute the epression for into the other equation. -8 + 0 + = 0 Distribute. - 5 = - 0 Combine like terms. = 6 Solve for. Step + (6) = 0 Substitute the value into either equation. + = 0 Simplif. = - Solve for. Step (-) + (6) 0 Check the solution in the other equation. -8 + 8 0 Simplif. 0 = 0 The solution is (-, 6). Eercises Solve each sstem b substitution. - - =. e. e a + b = - + = 5 a = -. e -m + n = 6-7m + 6n = 7 - = -. e + = = 9, = 7 a =, b = m = 7, n = 8 =, = 5 Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
Reteaching (continued) Solving Sstems Algebraicall Follow these steps when solving b elimination. Step Step Step Step Step 5 Step 6 Arrange the equations with like terms in columns. Circle the like terms for which ou want to obtain coefficients that are opposites. Multipl each term of one or both equations b an appropriate number. Add the equations. Solve for the remaining variable. Substitute the value obtained in step into either of the original equations, and solve for the other variable. Check the solution in the other original equation. + 5 = - What is the solution of the sstem of equations? e - = - Step + 5 = - Circle the terms that ou want to make opposite. - = - Step 6 + 5 = Multipl each term of the first equation b. -6 + = Multipl each term of the second equation b -. Step 9 = 57 Add the equations. Step = Solve for the remaining variable. Step 5 - () = - Substitute for to solve for. = - Step 6 (-) + 5() Check using the other equation. - + 5 = The solution is (-, ). You can also check the solution b using a graphing calculator. Eercises Solve each sstem b elimination. + = -7 5. e - = -9 6. e -5f + m = -6 -f - m = - 7. e - - = 5-6 + = 7 8. e - - = - 0 + 0 = -0 =, = f =, m = no solution = X, where is an real number 9. Reasoning Wh does a sstem with no solution represent parallel lines? If there is no solution, then there are no values of the variables that will make both equations true. This means there is no point that lies on both lines, so the lines never meet and are therefore parallel. Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
Reteaching Sstems of Inequalities Solving a Sstem b Using a Table An English class has computers for at most 8 students. Students can either use the computers in groups to research Shakespeare or to watch an online performance of Macbeth. Each research group must have students and each performance group must have 5 students. In how man was can ou set up the computer groups? Step Relate the unknowns and define them with variables. = number of research groups, = number of performance groups number of research groups + number of performance groups # number of research groups + 5 # number of performance groups 8 Step Make a table of values for and that satisf the first inequalit. The replacement values for and must be whole numbers. Step In the table, check each pair of values to see which satisf the other inequalit. Highlight these pairs. These are the solutions of the sstem. You can have: 0 groups doing research and 0,,, or groups watching performances or group doing research and 0,, or groups watching performances or groups doing research and 0,, or groups watching performances or groups doing research and 0 or group watching performances or groups doing research and 0 groups watching performances + + 5 8 0,,,, 0,,, 0,, 0, 0 0 0,,,, 0,,, 0,, 0, 0 0 Eercises Find the whole number solutions of each sstem using tables.. e + 6 + 0 6 - Ú. e 6 +. e + Ú 5 6 - + 8. e 6 + 6 (0, 0), (0, ), (0, ), (0, ), (, 0), (, ), (, ), (, 0), (, ), (, 0) (, 0), (, 0), (, ), (, 0), (, ) (0, 5), (0, 6), (0, 7), (, ), (, 5), (, ) (0, 0), (0, ), (0, ), (, 0), (, ), (, ), (, 0), (, ) Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
Reteaching (continued) Sstems of Inequalities Solving a Sstem b Graphing What is the solution of the sstem of inequalities? e - 7 + Ú Step Solve each inequalit for. - 7 + Ú - 7 - + and Ú - + 6 - Step Graph the boundar lines. Use a solid line for Ú or inequalities. Use a dotted line for 7 and 6 inequalities. O Step Shade on the appropriate side of each boundar line. The overlap is the solution to the sstem. Eercises O Solve each sstem of inequalities b graphing. 5. e 6. e + 7 Ú - - 6 O O 7. e 7 6 + O + 9 8. e - 7 O 9. e 6 - - Ú + O + 0. e + - O. e 7 - O. e + 6-6 O. e Ú - + O Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
Reteaching Sstems With Three Variables What is the solution of the sstem? + + z = -6 - + z = -9 - + + z = -9 Use elimination. The equations are numbered to make the process eas to follow. + + z = -6 - + + z = -9 + z = -5 - + z = -9 - + + z = -9 - + z = -9 - + + z = -8 5 + 7z = -7 + (-z) = -5 Equations and 5 form a sstem. 5- + (-7z) = -7 -z) = - Multipl equation 5 b - and add to equation to eliminate and solve for z. z) = - + () = -5 = -6 = - + + = -6 = - Substitute z = into equation and solve for. Substitute the values of and z into one of the original equations. Solve for. The solution is (,, ). Eercises Pair the equations to eliminate. Pair a different set of equations. Multipl equation b to eliminate. Then add the two equations. Solve each sstem b elimination. Check our answers. - + z = 0. + - 5z = 0 - + 5z = 8 - + z = 6. + + z = + 5 + z = 6 - + 5z =. 5 + + z = 5 + + 5z = (,, ) (,, ) (,, ). + + z = - + z = - + + z = 8 5. (,, ) (,, ) 5 + + z = 5 - - z = 9 + + z = 6 + + z = - 6. + + z = -0 - - z = -6 (,, ) Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
Reteaching (continued) Sstems With Three Variables What is the solution of the sstem? Use substitution. - + - z = 9 - - + z = 8 - + + z = 5 Step Choose an equation that can be solved easil for one variable. Choose equation and solve for. + - z = 9 = - + z + 9 Step Substitute the epression for into equations and and simplif. (- + z + 9) - + z = 8 -(- + z + 9) + + z = 5 - + 8z + 76 - + z = 8 9-6z - 57 + + z = 5 - + z = -68 5 - z = 7 Step Write the two new equations as a sstem. Solve for and z. e 5 - + z = -68 - z = 7-56 + z = -7 Multipl b. - + z = -68 Substitute = 8 into. - z = 79 Multipl 5 b. -(8) + z = -68 65 = 50 z = = 8 z = Step Use one of the original equations to solve for. + - z = 9 Substitute = 8 and z = into. + (8) - () = 9 = The solution of the sstem is (, 8, ). Eercises Solve each sstem b substitution. Check our answers. - + - z = 7 7. - - + z = 5 - + - z = -6 (, 7, ) - + - z = - 0. - + + z = - - + z = (,, ) - - + 5z = -5 8. - + - 7z = 0 - - + 6z = - (,, ). - + z = 7-5 - z = - + - 6z = - (6,, ) 9.. - + z = 5 - - z = 7 + + 5z = (5,, ) - - - z = - - + - 5z = -5-5 + + 5z = (,, 5) Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
Reteaching Solving Sstems Using Matrices How can ou represent the sstem of equations with a matri? c - - + 5z = - - + = - + + z = 7 Step Write each equation in the same variable order. Line up the like variables. Write in variables that have a coefficient of 0. c - - + 5z = - - + + 0z = - - + + z = -7 Step Write the matri using the coefficients and constants. Remember to enter a for variables with no numeric coefficient. D - 5 0 - - T 7 Eercises Write a matri to represent each sstem. + = -. e = 5 c 0 ` 5 d. c - 5 + z = 9 + 7 + z = - z = 5 D 7 0 9 T. c 5 - + z = + z = 6 + + z = 5 D 0 6 T. c - z = 5 + z = -5 - + = 5. c z = 6 + = - - 5z = 0 6. c z - 5 + = - + + z = - + z - = -5 0 D 0 5 0 5 T 0 0 D 0 5 6 T 0 5 D T 5 Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
Reteaching (continued) Solving Sstems Using Matrices What is the solution of the sstem? e + 5 = 5 - + = -7 Step Write the matri for the sstem. c 5 - ` 5-7 d Step Multipl Row b. Add to Row. Replace Row with the sum. Write the new matri. 5 5) +(- -7) 0 9-9) c 0 9 - ` -9-7 d Step Divide Row b 9. Write the new matri. 9 (0 9-9) c 0 - ` - -7 d Step Multipl Row b -. Add to Row. Replace Row with the sum. Write the new matri. -(0 -) + - -7) - 0-5) c 0-0 ` - -5 d Step 5 Multipl Row b -. Write the new matri. -(- 0-5) c 0 0 ` - 5 d This matri is equivalent to the sstem e = -. The solution is (5, -). = 5 Eercises Solve each sstem of equations using a matri. - + = 6 7. e - - = - (, ) 6 + = - 8. e - + = + = - 9. e - - = 7 (, ) (, 5) Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.