Grade 8 / Equations 81 exercises (100% corrected)

a
image PDF : /home/_math/_exercice/d0/737/metapost/dessin-1.pdf
image PDF : /home/_math/_exercice/d0/737/metapost/dessin-2.pdf
image PDF : /home/_math/_exercice/d0/737/metapost/dessin-3.pdf
image PDF : /home/_math/_exercice/d0/737/metapost/dessin-4.pdf
image PDF : /home/_math/_exercice/d0/737/metapost/dessin-5.pdf
image PDF : /home/_math/_exercice/d0/737/metapost/dessin-6.pdf
image PDF : /home/_math/_exercice/d0/737/metapost/dessin-7.pdf
xxxx3 5x7 x3×x3×x511×3 ChingQuizz : 11 exercises available for Quizz assessment : 1. Reminders E.8986 Which of the numbers 4 or 5 is a solution of the equation : 5 x + 1 = 3 x + 11 E.9053 Without justification and in each case, give the value of x to balance the balance : a b E.9054 We wish to solve the equation 3 x +5=11 . 1 Complete the following diagram: 2 A student’s essay is tran-scribed below with passages deleted. Copy and complete this essay: E.1122 Solve the following equations : a 3 x + 2 = 5 b 3.6 x − 0.6 = − 6 2. Equality and balance E.737 Shown below are the manipulations performed on a balance to determine the unknown value x . https://chingmath.fr chapExoCorrec/8986 sacados/8986 chapExoCorrec/9053 sacados/9053 xxxx3 5x7 chapExoCorrec/9054 sacados/9054 x3×x3×x511×3 chapExoCorrec/1122 sacados/1122 chapExoCorrec/737 sacados/737
x212x 72x3x3 x41xx x43x6x 4x2x143x2143x12x4 2x3x4 7x32x13 =5x23x10Soustraire3xauxdeuxmembres=Soustraire2auxdeuxmembres=Diviserpar2lesdeuxmembres= 1 For each step, indicate the manipulation that was on each of the scales. 2 Deduce the value of the unknown x . E.697 Without justification and in each case, give the value of x allowing the balance to be balanced : a b E.4884 Without justification and in each case, give the value of x to balance the balance : a b 3. Equations with positive solutions E.9032 An unknown value balances the two pans of the scale below : Successive modifications have been applied to the two pans of the balance to ensure that it remains balanced at all timesand the last ˇ state ı of the balance shows that the unknown value was 4 . Complete the ˇ bubbles ı explaining the various actions that were performed on the balance. E.4903 Determine, for each question, the value of x achieving the balance of the scale : a b E.9046 We wish to know the solutions of the equation : 5 x + 2 = 3 x + 10 To solve this equation and by the principle of equilibrium, we’ll apply the algebraic operations below : Deduce the solution to this equation. E.329 We wish to solve the equation : 3 x + 2 = x + 8 1 Complete the wording below : 3 x + 2 = x + 8 3 x + 2 − . . . = x + 8 − . . . 3 x = x + . . . 3 x − . . . = x + . . . − . . . . . . x = . . . . . . x . . . = . . . . . . x = . . . 2 Give the solution to this equation. https://chingmath.fr chapExoCorrec/697 sacados/697 x212x 72x3x3 chapExoCorrec/4884 sacados/4884 x41xx x43x6x chapExoCorrec/9032 sacados/9032 4x2x143x2143x12x4 chapExoCorrec/4903 sacados/4903 2x3x4 7x32x13 chapExoCorrec/9046 sacados/9046 =5x23x10Soustraire3xauxdeuxmembres=Soustraire2auxdeuxmembres=Diviserpar2lesdeuxmembres= chapExoCorrec/329 sacados/329
=3x−55x11Soustraire5xauxdeuxmembres=Ajouter5auxdeuxmembres=Diviserpar−2lesdeuxmembres= E.9052 We wish to solve the equation : 8 x + 1 = 5 x + 4 1 Complete the essay below : 8 x + 1 = 5 x + 4 8 x + 1 − . . . = 5 x + 4 − . . . 8 x = 5 x + . . . 8 x − . . . = 5 x + . . . − . . . . . . x = . . . . . . x . . . = . . . . . . x = . . . 2 Give the solution to this equation. E.9033 Solve the following equations : a 3 x + 7 = x + 13 b 8 x + 2 = 2 x + 20 E.8979 Solve the following equations : a 6 x + 1 = x + 4 b 7 x + 2 = 3 x + 7 E.1996 Solve the following equations : a 12 x + 5 = 2 x + 12 b 6 x = 4 x + 13 E.4896 Solve the equations below. a 5 x + 0.2 = 2 x + 5 b 2 x + 0.4 = 0.8 x + 4 E.4897 Solve the equations below. Results will be given as simplified fractions. a 8 x + 2 = 2 x + 17 b 3 x + 3 = x + 8 E.9034 Solve the equations below. Results will be given as simplified fractions. a 8 x + 5 = 4 x + 11 b 10 x + 5 = 4 x + 20 E.9035 Solve the equations below. Results will be given as simplified fractions. a 6 x + 7 = 2 x + 9 b 7 x + 5 = 3 x + 19 4. Equations using relative numbers E.9051 We wish to know the solutions of the equation : 5 x + 2 = 3 x + 10 To solve this equation and by the principle of equilibrium, we’ll apply the algebraic operations below : Deduce the solution to this equation. E.9039 Solve the following equations : a 5 x + 3 = 2 x − 3 b 5 x − 3 = 3 x − 5 E.9038 Solve the following equations : a 4 x − 5 = 2 x − 7 b 2 x − 4 = 5 x + 8 E.4900 Solve the following equations : a 3 x + 4 = 8 x − 21 b 2 x − 16 = 5 x + 2 E.4894 Solve the following equations : a 3 x + 5 = 5 x + 8 b 6 x − 2 = − x − 6 E.1111 Solve the following equations : a 2 x + 5 = 5 x − 4 b − 8 x + 2 = 3 x − 8 E.1115 Solve the following equations : a − 3 x + 5 = 2 x − 20 b − 2 x + 1 = 7 x − 80 E.9040 Solve the following equations : a 2 x + 3 = − 4 x b x + 2 = 2 − x E.9036 Solve the following equations : a − 5 − 3 x = − 2 x + 13 b − 8 x − 3 = − 3 x − 6 E.9041 Solve the following equations and give the results as a reduced fraction : a 3 − 4 x = 10 · x + 7 b − 3 x + 2 = x + 4 E.9037 Solve the following equations and give the results in reduced fraction form : a − 3 x + 5 = 3 x − 16 b 2 − 3 x = 5 x + 6 5. Equation and operations on fractional numbers https://chingmath.fr chapExoCorrec/9052 sacados/9052 chapExoCorrec/9033 sacados/9033 chapExoCorrec/8979 sacados/8979 chapExoCorrec/1996 sacados/1996 chapExoCorrec/4896 sacados/4896 chapExoCorrec/4897 sacados/4897 chapExoCorrec/9034 sacados/9034 chapExoCorrec/9035 sacados/9035 chapExoCorrec/9051 sacados/9051 =3x−55x11Soustraire5xauxdeuxmembres=Ajouter5auxdeuxmembres=Diviserpar−2lesdeuxmembres= chapExoCorrec/9039 sacados/9039 chapExoCorrec/9038 sacados/9038 chapExoCorrec/4900 sacados/4900 chapExoCorrec/4894 sacados/4894 chapExoCorrec/1111 sacados/1111 chapExoCorrec/1115 sacados/1115 chapExoCorrec/9040 sacados/9040 chapExoCorrec/9036 sacados/9036 chapExoCorrec/9041 sacados/9041 chapExoCorrec/9037 sacados/9037
E.9063 Solve the following equation : 5 3 x + 3 = x − 2 E.9047 Solve the following equations : a 4 3 x − 1 = 2 3 x + 1 6 b 3 5 x + 7 5 = 1 5 x + 1 5 E.9048 Solve the following equations : a − 5 3 x + 3 5 = 1 6 x − 1 2 b 1 14 x + 2 = 4 7 x + 1 4 6. Simple distributivity and equations E.1112 Solve the following equations : a 3 x + 1 = 3(2 − x ) b 4(5 − 2 x ) = 3 x E.1998 Solve the equations : a 2( x − 2) = 3(5 − 2 x ) b 3( x + 1) = 2(3 x − 5) E.4905 Solve the following equations : a 2( x + 3) = 4( x − 1) b 5(1 − x ) = 3(2 x + 1) E.1118 Solve the following equations : a 2( x + 4) = 3(2 − x ) b 4( x + 2) = 4(2 x + 3) E.1116 Solve the following equations : a 2( x − 2) = 3 x + 3(2 x + 1) b 3(2 − 3 x ) + 4( x + 2) = 4 x + 2( x − 2) 7. Simple distributivity, equations, and multiplication of fractions E.9042 Solve the following equations : c − (3 − x ) = 5 x + 2 d − 3(2 x − 1) = x + 2 E.9043 Solve the following equations : a − ( x − 2) = 2(2 x + 1) b − ( x + 2) = 3(2 x + 1) E.6419 Solve the following equations : a 2 x + 1 − 4 = 4 x + 1 b 3 − 2 2 x + 1 = 5 2 − x E.9045 Solve the following equations : a 2( x − 1) − 3(2 x − 4) = 3 x + 5 b 5(5 − 3 x ) − 2( x + 1) = 3 x + 6 E.9044 Solve the following equations : a − 3( x + 2) + 4(5 − x ) = 2 x + 5 b 4(2 x + 4) − 3(5 − x ) = 2( x + 1) E.4912 Solve the following equations : a 2( x + 1) − 3(2 x − 4) = − 3( − x + 1) b 5 x + 2 − 3(2 − 4 x ) = 2(3 x + 4) 8. Double distributivity and equations E.8980 Solve the equation : 4 x +3 3 x − 2 = 2 x − 1 6 x +4 E.9049 Solve the following equation : 3 x − 2 3 x + 4 − 2 x + 1 4 x + 1 = 0 E.9050 Solve the equation : 2 x − 6 3 x + 4 − x − 3 2 x − 4 = 0 E.9129 Solve the following equation : 4 x − 3 4 x + 4 = 3 x − 1 5 x + 3 9. Problems: equality of expressions https://chingmath.fr chapExoCorrec/9063 sacados/9063 chapExoCorrec/9047 sacados/9047 chapExoCorrec/9048 sacados/9048 chapExoCorrec/1112 sacados/1112 chapExoCorrec/1998 sacados/1998 ##5225 chapExoCorrec/4905 sacados/4905 chapExoCorrec/1118 sacados/1118 chapExoCorrec/1116 sacados/1116 chapExoCorrec/9042 sacados/9042 chapExoCorrec/9043 sacados/9043 chapExoCorrec/6419 sacados/6419 chapExoCorrec/9045 sacados/9045 chapExoCorrec/9044 sacados/9044 chapExoCorrec/4912 sacados/4912 chapExoCorrec/8980 sacados/8980 chapExoCorrec/9049 sacados/9049 chapExoCorrec/9050 sacados/9050 ##5225 chapExoCorrec/9129 sacados/9129
ABCx5x3 Cube dont l’arêtemesure8petits cubesCube dont l’arêtemesure9petits cubes ABC3x−2x4DEFG2x−13x−5 E.1341 Henry is 6 times the age of his daugh-ter Annette and the sum of their ages is 42 . 1 Noting x Annette’s age, only one of the equalities below is true. Which one? a x = 42 ÷ 6 b 6 x = 42 + x c 6 x + x = 42 d 6 + x = 42 2 Without justification, determine Annette’s age. E.2469 Today, Mark is 11 years old and Pe-ter is 26. In how many years will Peter’s age be twice that of Mark? The approach taken will be detailed on the copy. E.5773 A zoo is preparing to receive ani-mals in two cages of equal weight. The composition of each of these cages is as follows : the first cage: a goat and two dolphins ; the second cage: an elephant and a dolphin ; The elephant weighs 130 kg , the goat weighs 60 kg , and all the dolphins weigh the same. We note x the weight of a dolphin. 1 Determine an expression as a function of x giving the weight of the left cage? And also of the one on the right? 2 What is the weight of a dolphin? E.1336 Three years ago Cecile was a third of her father’s age. Her father is now 39 years old. Determine Cecile’s age. E.5775 A zoo is preparing to receive animals in two of the same weight transport cages. The composition of each of these cages is as follows : first cage: one goat and three dolphins ; second cage: an elephant calf, a goat, and a dolphin ; The elephant weighs 132 kg , the goat weighs 55 kg , and all the dolphins weigh the same. What is the weight of a dolphin? E.4925 1 Give the expanded and reduced form of the expression : ( x + 3)( x + 3) . 2 Let x be a positive number. Consider the triangle ABC whose measures are: AB = x ; BC =5 ; AC = x +3 Determine the value of x so that the triangle ABC is rectangular to B . E.5774 Adrian had lunch at the same restau-rant on two consecutive days. Here are those two notes : Plate of deli . . . . . . . . . . . . . . . . . . . 9 e Fried ground steak . . 12 e 3 Coffees . . . . . . . . . . . . xxx Total . . . . . . . . . . . . . . . . . xxx Goat cheese salad chaud. . . . . . . . . . . . . . 11.5 e Escalope of veau . . . . 14 e 1 Café . . . . . . . . . . . . . . . xxx Total . . . . . . . . . . . . . . . . . xxx Some of the information in these two notes has faded, but we know that he paid the same amount for both meals. Find the price of a coffee at this restaurant. E.3893 Subtracting 3 from a num-ber or dividing it by 3 gives the same result. What is that number? Justify your answer. E.6355 Here is a representation of the ˇskeletonı of cubes constructed from small cubes. Following this construction scheme, Jane used 140 small cubes. Of how many small cubes is the edge of the large cube made up of? 10. Problems, areas and volumes E.9055 We consider the two geometric fig-ures below : https://chingmath.fr chapExoCorrec/1341 sacados/1341 chapExoCorrec/2469 sacados/2469 chapExoCorrec/5773 sacados/5773 chapExoCorrec/1336 sacados/1336 chapExoCorrec/5775 sacados/5775 chapExoCorrec/4925 sacados/4925 ABCx5x3 chapExoCorrec/5774 sacados/5774 chapExoCorrec/3893 sacados/3893 Antilles Guyane Septembre 2009 chapExoCorrec/6355 sacados/6355 ##5225 Cube dont l’arêtemesure8petits cubesCube dont l’arêtemesure9petits cubes chapExoCorrec/9055 sacados/9055 ABC3x−2x4DEFG2x−13x−5
xx3cm4cmABCDEFGHIJxx ABCDFx46 26cm 20m12m ABCEM Determine the value x so that the triangle and rectangle have the same perimeter. E.9056 Consider the two polygons shown below : où x is an indeterminate measure measured in centimeters and où : polygon ABCDEF consists of a square of side x and a rectangle of dimensions 4 cm and 3 cm . polygon GHIJ is a square of side x +2 . Determine the value of x so that the polygons ABCDEF and GHIJ have the same area. E.3927 Consider the figure below o where dimensions are given in centimeters and areas in cm 2 . ABCD is a rectangle. The triangle DCF is a rectangle D 1 Show that the area of rectangle ABCD is : 24 − 4 x 2 Show that the area of the triangle DCF is : 2 x . 3 Determine the value of x so that the rectangle ABCD and the triangle DCF have the same area. E.5772 In a square of 26 cm side, we cut out a small square. The area re-maining and shown hatched below has area 576 cm 2 . Determine the length of the side of the cut-out square. E.5771 A swimming pool has length 20 m and width 12 m . A wooden walkway is desired around the perimeter of the pool as shown in the sketch below : Knowing that the outside perimeter of the wooden walkway has measure 76 m , determine the width of the wooden sur-round. E.3910 Consider a triangle ABC right-angled A such that : AB = 6 cm ; AC = 4 cm M is a point on segment [ AB ] . The line through the point M and perpendicular to the line ( AB ) intersects the segment [ BC ] at E . We wish to place the point M on the segment [ AB ] so that the triangle AEM is isosceles at M . We pose : x = BM 1 Show that the distance EM is written as a function of M : EM = 2 3 · x 2 Deduce the position of M on segment [ AB ] so that AEM is isosceles at M . E.6412 Three identical equilateral triangles are cut from the corners of an equilateral triangle of side 6 cm . The sum of the perimeters of the three small triangles is equal to the perimeter of the remaining gray hexagon. What is the measure of the side of the small triangles? Any trace of research, even unsuccessful, will appear on the copy and will be taken into account in the scoring. https://chingmath.fr chapExoCorrec/9056 sacados/9056 xx3cm4cmABCDEFGHIJxx chapExoCorrec/3927 sacados/3927 ABCDFx46 chapExoCorrec/5772 sacados/5772 26cm chapExoCorrec/5771 sacados/5771 20m12m chapExoCorrec/3910 sacados/3910 ABCEM chapExoCorrec/6412 sacados/6412
6x24x3 ABCDE7x5x3;5 E.5781 Underneath a staircase is a storage room whose shape is a right prism with a right triangle as its base. Its dimensions are: Length: 7 m Height: 3 m Depth: 2 m We wish to place the largest possible cube in this storage room. The diagram below is given as an example: What is the volume of the largest cube that can be placed in this storage room? 11. Share E.11338 Solve the following three equations : a 3 x + 5 = 9 x − 10 b 3(2 x + 1) = 3 x + 2 c 4( x + 1) − 2(5 x − 5) = 0 12. Unclassified exercises E.5778 The adjacent figure shows a bal-ance in equilibrium: the left and right pans have the same weight. On each tray are weights whose mass is indicated on their front face où x represents the same positive number on both trays. What is the value of the number x achieving this equilibrium situation? E.5777 Yuna and Peter each have a calculator. They ˇ typed ı the same number. Then Yuna pressed the keys : × 4 + 3 = and Peter pressed the keys : − 1 = × 7 + 8 = Surprise! They also get the same result! What number could they have chosen? E.5776 Emma and Zoe each have a calculator. They ˇ typed ı the same number. Next, Emma pressed the keys : × 4 + 3 = and Zoe pressed the keys : − 1 = × 2 + 8 = Surprise! They get the same result! What number could they have chosen? E.4480 Solve the following equation : 3(2 x − 5) − 2(1 − x ) = 2 x − 7 E.9212 Using cross multiplication, solve the following equations : a x + 1 2 = 3 5 b 2 x + 1 3 = 1 2 c 2 x + 1 x − 1 = 1 5 E.9213 Solve the following equations : a 3 2 + x = 5 4 b 1 − x 2 = 5 3 E.6387 Consider the figure below : Without justification, give the length of segment [ DE ] . Hint : we will expand the product ( x +5)( x +5) . E.5259 Solve the following equations : a x + 1 2 x + 1 = 3 x 6 x + 1 b x 2 x + 1 = 3 − 2 x − 4 x E.5660 Using the cross product, solve the following equations : a x 3 = 3 2 b x 7 = 2 49 c x 5 = 3 4 E.831 Solve the following equations : a x 2 = 2 3 b 3 x = 7 2 c 4 = 6 x d 15 x 12 = 25 4 E.1113 Solve the following equations using the cross product : a 2 x 5 = 3 7 b 2 7 = 3 x c 3 x 14 = 6 7 https://chingmath.fr chapExoCorrec/5781 sacados/5781 chapExoCorrec/11338 sacados/11338 chapExoCorrec/5778 sacados/5778 6x24x3 chapExoCorrec/5777 sacados/5777 chapExoCorrec/5776 sacados/5776 chapExoCorrec/4480 sacados/4480 chapExoCorrec/9212 sacados/9212 chapExoCorrec/9213 sacados/9213 chapExoCorrec/6387 sacados/6387 ABCDE7x5x3;5 chapExoCorrec/5259 sacados/5259 chapExoCorrec/5660 sacados/5660 chapExoCorrec/831 sacados/831 chapExoCorrec/1113 sacados/1113
E.1121 Solve the following equations using the cross product : a x x + 1 = 3 2 b 2 x + 1 3 x − 2 = 5 7 c 2 x − 3 3 = 3 − 2 x 7 E.463 Solve the following equations : a x − 1 = 3 2 b 1 2 x − 1 = 0 c x + 1 = 2 x − 1 d 2( x − 1) − 4(2 − x ) = 3 x − 7 e x 2 + x + 1 = ( x + 1)( x − 1) E.4895 Solve the following equations : a 5 x + 7 = 3 x + 13 b 2 x + 1 = x + 8 c 9 x + 6 = 5 x + 14 d 4 x + 1 = x + 10 E.11783 Résoudre les équations suivantes : a 5 x + 7 = 3 x + 13 b 2 x + 1 = x + 8 https://chingmath.fr chapExoCorrec/1121 sacados/1121 chapExoCorrec/463 sacados/463 chapExoCorrec/4895 sacados/4895 chapExoCorrec/11783 sacados/11783