Grade 8 / Introduction to equations 54 exercises (100% corrected)

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ChingQuizz : 14 exercises available for Quizz assessment : 1. Status of equality E.8983 Definition: An equality defines two objects one on the left ( left member) , the other on the right ( right member) of the sign ˇ = ı. A numerical equality is true if the left-hand member has the same value as the right-hand member. Other-wise it is said false . Among the equalities below, say whether they are true or false : a 3 + 2 = 6 b 3 − 3 × 5 = 2 × ( − 6) c 5 × 2 = 12 − 2 d 2 × ( − 3 + 2) = 5 − 2 × 3 E.8999 Of the equalities below, say whether they are true or false : a 3 × 2 5 − 1 5 = 3 2 × 4 3 − 1 b 5 6 2 = 1 12 + 11 18 E.9000 Of the equalities below, say whether they are true or false : a 12 2 +9 2 = 25 2 − 20 2 b 7+ 5 − 1 = 5 2 − 3 2. Solutions of an equation E.1975 Definition: on the equality of literal expressions The equality ˇ 3 x +2 =3 x +6 ı is true for any value of x : we speak of identité . The equality ˇ 3 x +2=20 − x ı is false because evaluated in 1 , it proposes the equality 5=19 . We then speak of equation which can take the status true or false depending on the value of x . For the value 4 , the equation ˇ 3 x +2=18 − x ı is true : we say that 4 is a solution of this equation. Consider the equation : ( E ): 3 x +7= x +11 1 Evaluate the two expressions 3 x +7 and x +11 for the value x =1 . Does the number 1 verify the equality of the equation ( E ) ? 2 Evaluate the two expressions 3 x +7 and x +11 for the value x =2 . Does the number 2 verify the equality of the equation ( E ) ? E.7932 For which equations the number 3 is a solution. a 3 x + 2 = 2 x − 4 b 2( x + 4) = 3 x + 5 E.9001 Consider the equation : ( F ) : 2 x + 3 = 3 x + 1 . Which of the following numbers is the solution to this equa-tion : A 1 b − 1 c 2 d 5 E.9002 Consider the equation : ( G ) : 5 x − 1 = − x − 7 . Which of the following numbers is the solution to this equa-tion : A 1 b 2 c 0 d − 1 E.1328 Each of the following equations ad-mits at least one solution from among the numbers − 2 , − 1 , 0 , 1 and 2 . Associate each of these equations with its solution. a 2 × x + 3 = 4 × x + 1 b 2 − x = 4 + x c 3 × ( x + 2) = 6 × x d ( x + 2) × ( x + 3) = 0 E.1123 For which equations is the number 3 a solution? a x + 3 2 × x = 5 3 − 2 x b 3 − x x = 5 − 3 × x E.9003 Consider the equation ( E ) defined by: x 2 + 3 x + 1 = 3 − x 2 Is the number − 2 a solution of the equation ( E ) ? Justify. 3. Use of inverse operations https://chingmath.fr chapExoCorrec/8983 sacados/8983 chapExoCorrec/8999 sacados/8999 chapExoCorrec/9000 sacados/9000 chapExoCorrec/1975 sacados/1975 chapExoCorrec/7932 sacados/7932 chapExoCorrec/9001 sacados/9001 chapExoCorrec/9002 sacados/9002 chapExoCorrec/1328 sacados/1328 chapExoCorrec/1123 sacados/1123 chapExoCorrec/9003 sacados/9003
??5×2??8−5??−3 ??−4×6??9÷2??7;5 ??3×7??28÷9??1;1 −2;6??1;4×1;5??6÷2;5??2 ??73×43??73÷253??35 x2×x2×x1763×2÷2−1 E.9004 On each of the diagrams below, com-plete the missing information : 1 Complete the following diagrams : 2 Give the solutions to each of the following equations : a x + 3 = 5 b 2 x = 8 c x − 5 = − 3 d x + 3 = − 4 e 6 x = 9 f x ÷ 2 = 7.5 E.1342 1 Complete each of the following diagrams : 2 Solve the following equations : a x + 10 = 3 b 7 x = 28 c x ÷ 9 = 1.1 E.1311 Solve the following equations : a x + 5 = 12 b x + 7 = 3 a x − 3 = 2 b x − 6 = − 2 E.1322 Solve the following equations : a 5 × x = 35 b 3 × x = 15 c x ÷ 4 = 5 d x ÷ 3 = 9 E.9005 1 Complete each of the following diagrams : 2 Solve the following equations : a x + 2.6 = 1.4 b 1.5 x = 6 c x ÷ 2.5 = 2 E.1309 Solve the equations : a x + 2.5 = 5.1 b x + 3.4 = 6.3 c x − 3.1 = 3.2 d x − 4 = − 4 E.1310 Solve the following equations : a x + 5.2 = 7.3 b x + 2.1 = 4 c x − 2.1 = 3.4 d x − 4.9 = 2.3 E.1323 Solve the following equations : a x − 2.5 = 6 b x − 5.2 = − 2 c 2 + x = 5 d 2.5 + x = 6.3 E.1981 1 Complete each of the following diagrams : 2 Solve the following equations : a x + 2 = 7 3 b 4 3 × x = 7 3 c x ÷ 25 3 = 3 5 E.1318 Solve the following equations and express the solutions as simplified fractions : a 6 × x = 8 b 12 × x = 20 c 1.2 × x = 5.4 d 2.4 × x = 1.8 4. Special cases E.8984 1 What are the solution numbers of the equation : 0 · x =0 2 What are the solution numbers of the equation : 0 · x =3 5. Arithmetic solutions of equations of the form ax + b = c E.8990 Note: for the equation 2 x +1=7 , finding solutions is easy because, with the help of operation priorities, all you need to do is see how the expression of the left-hand member is constructed to obtain the numerical value of the right-hand member. The diagram below shows this resolution : The solution to the equation 2 x +1=7 is the number 3 . https://chingmath.fr chapExoCorrec/9004 sacados/9004 ??5×2??8−5??−3 ??−4×6??9÷2??7;5 chapExoCorrec/1342 sacados/1342 ??3×7??28÷9??1;1 chapExoCorrec/1311 sacados/1311 chapExoCorrec/1322 sacados/1322 chapExoCorrec/9005 sacados/9005 −2;6??1;4×1;5??6÷2;5??2 chapExoCorrec/1309 sacados/1309 chapExoCorrec/1310 sacados/1310 chapExoCorrec/1323 sacados/1323 chapExoCorrec/1981 sacados/1981 ??73×43??73÷253??35 chapExoCorrec/1318 sacados/1318 chapExoCorrec/8984 sacados/8984 chapExoCorrec/8990 sacados/8990 x2×x2×x1763×2÷2−1
x17 x2x2x−31×2−3 x3×x3×x511 x6×x6×x−210 Determine the solution to the equation 7 x +3=17 by complet-ing the diagram below : E.5254 Without justification, give the solu-tions of the following equations : a 3 x + 1 = 4 b 5 x − 4 = 6 c 2 × x + 1 = 5 E.1983 Without justification, give the solu-tions of the equations below : a 5 × x + 10 = 16 b − 2 × x + 1 = 5 E.5249 We consider the following calcula-tion program : Choose a starting number; Multiply the number by 2 ; Subtract 3 ; Write the final result. 1 When the number chosen as the input to the calculation program is 5 , give the output number of this calculation program. 2 When choosing an input number, the calculation pro-gram returns the number 1 . This is illustrated in the diagram below : What is the number chosen as input to the calculation program. E.8991 Without justification, give the solu-tions of the equations below : a − 2 − x = 5 b 2 − 3 x = 2 6. Writing equations of the form ax + b = c E.1982 We wish to solve the equation 3 x +5=11 . 1 Complete the following diagram: 2 A student’s essay is transcribed below with passages deleted. Copy and complete this essay: E.8987 We wish to solve the equation : 6 x − 2=10 . 1 Complete the following diagram: 2 A student’s essay is transcribed below with passages deleted. Copy and complete this essay: E.7994 Solve the following equations : c 2 x + 1 = 2 b 10 × x + 9 = 10 Hint: Give the solutions in decimal form. E.11360 Solve the equations : a 4 x + 3 = 16 b 3 x − 8 = 13 c 5 x + 8 = 17 Hint: Give the solutions in decimal form. E.11361 Solve the equations : a − 2 x + 6 = 11 b − 3 x + 14 = − 7 c − 5 x + 11 = 5 Hint: Give the solutions in decimal form. E.8989 Solve the following equations and give the results in simplified fraction form : a 3 × x + 1 = 5 b 15 × x + 4 = − 1 Hint: Give the solutions in decimal form. E.11362 Solve the equations : a 6 x − 7 = 7 b 9 x − 4 = − 1 c 12 x + 5 = 15 Hint: Give the solution in the form of a reduced fraction. E.11396 Solve the following equations : a 10 × x + 1 = 3 b 4 × x + 12 = 6 Hint: Give the solutions in decimal form. https://chingmath.fr x17 chapExoCorrec/5254 sacados/5254 chapExoCorrec/1983 sacados/1983 chapExoCorrec/5249 sacados/5249 x2x2x−31×2−3 chapExoCorrec/8991 sacados/8991 chapExoCorrec/1982 sacados/1982 x3×x3×x511 chapExoCorrec/8987 sacados/8987 x6×x6×x−210 chapExoCorrec/7994 sacados/7994 chapExoCorrec/11360 sacados/11360 chapExoCorrec/11361 sacados/11361 chapExoCorrec/8989 sacados/8989 chapExoCorrec/11362 sacados/11362 chapExoCorrec/11396 sacados/11396
Choisir un nombreAjouter 2Multiplier par4Multiplier par3Ajouter les deux nombres 3x4x8x5x E.5253 We consider the following calcula-tion program : Choose a starting number; Multiply the number by − 2 ; Add 3 ; Write the final result. 1 Writing x the starting number, give the literal expression obtained at the end of this calculation program. 2 Give the value for which this calculation program returns the value 5 . 7. Algebra and equations E.7993 The following calculation program is considered : 1 Noting x the number chosen as the input to this compu-tational program, give the algebraic expression obtained as the output of this computational program. 2 By solving an equation, determine the number chosen as input so that this computation program returns the value 13 . E.8988 Solve the following equations : a 2 × (3 x + 1) = 8 b − x + 5 + 4 x − 2 = 1 E.8993 Solve the following equations : a x + 3 2 x − 2 − 2 x 2 + 5 = 14 b − 2 × 6 x 2 + 3 − 4 x × 2 − 3 x = 0 E.8992 Solve the following equations : a 3 × 3 − 5 x + 15 x = 5 b 6 x × 3 x − 6 + 9 x + 9 2 − 2 x = − 18 8. Algebra, equations and problems E.1326 Jean buys 3 binders and 6 notebooks. The first piece of information Jean has is that a binder costs 12 pesos more than a notebook: 1 Noting x the price of a notebook, justify that the total price of 3 binders and 6 notebooks is expressed as : 9 x + 36 2 Knowing that John’s total purchases amount to 126 pe-sos, determine the price of a notebook and a binder. E.1988 A supermarket has just received 1 500 yoghurts : plain yoghurts and fruit yoghurts. We note x the number of fruit yogurts. 1 Express the number of plain yogurts as a function of x . 2 a Knowing that a plain yogurt costs 1.5 e et that a fruit yogurt costs 2.25 e , show that the total selling price of the 1 500 yogurts is expressed as : 0.75 x + 2250 b The sale of all these yogurts brought in 2 925 e . Deter-mine the number of plain yogurts and the number of fruit yogurts sold by the supermarket. E.4924 In a banana plantation, a farmer pro-duces two types of bananas : plantains and sweet bananas. To his distributors, he sells 4 e the plantain diet and 6 e le the sweet banana diet. With each load, the farmer’s truck carries 1 000 bunches of bananas, and today his load is charged 5 256 e . Noting x the number of plantain bunches contained in this load, determine the number of plantain bunches and sweet bunches comprising this load. E.1999 A school with 1 300 students decides to organize an event where all the students will be present. To this end, each student must buy a costume from the school: this costume costs 20 e for girls and 15 e pour for boys. For the sale of these costumes, the high school raised 23 100 e . Noting x the number of boys present in the establishment, determine the number of girls and boys making up this estab-lishment. 9. Algebra, equations, perimeters and areas E.1338 Consider the trapezoid below : https://chingmath.fr chapExoCorrec/5253 sacados/5253 chapExoCorrec/7993 sacados/7993 Choisir un nombreAjouter 2Multiplier par4Multiplier par3Ajouter les deux nombres chapExoCorrec/8988 sacados/8988 chapExoCorrec/8993 sacados/8993 chapExoCorrec/8992 sacados/8992 chapExoCorrec/1326 sacados/1326 chapExoCorrec/1988 sacados/1988 chapExoCorrec/4924 sacados/4924 chapExoCorrec/1999 sacados/1999 chapExoCorrec/1338 sacados/1338 3x4x8x5x
DEFG2x−13x−5 ABCD4cm2x−3 EFGH7cm3x1 ABC3x−2x14 4x311 5x1237 8x24x14 3x2x6 5x32x7 7x23x7 4x12x4 12x38x17 4x33x5 5x12x10 5x32x7 7x23x7 1 Write a literal expression expressing the perimeter of the trapezoid opposite 2 Determine the value of x so that the trapezoid has a perimeter of 44 m . E.8981 Consider the geometric figure opposite : Determine the value of x so that the rectangle DEFG has a perimeter of 10 cm . E.4907 Consider the geometric figure below : Determine the value of x so that the rectangle ABCD has area 20 cm 2 . Leave any trace of research even if it is not successful. E.8982 Consider the geometric figure oppo-site : Determine the value of x so that the rectangle EFGH has area 49 cm 2 . E.1976 1 A rectangle has a width of 4 cm and a perimeter measure 24 cm . Determine the measure of its length. 2 A rectangle has an area of 21 cm 2 and a length of 7 cm . Determine the measure of its width. 10. Unclassified exercises E.4906 Consider the triangle ABC below : Determine the value of x so that the triangle ABC has a perimeter of 9 cm . E.11779 Déterminer, pour chaque question, la valeur de x réalisant l’équilibre de la balance : a b E.375 For each question, determine the value of x that balances the scale : a b E.4893 For each question, determine the value of x that balances the scale : a b c d E.11777 Déterminer, pour chaque question, la valeur de x réalisant l’équilibre de la balance : a b E.11782 Déterminer, pour chaque question, la valeur de x réalisant l’équilibre de la balance : a b https://chingmath.fr chapExoCorrec/8981 sacados/8981 DEFG2x−13x−5 chapExoCorrec/4907 sacados/4907 ABCD4cm2x−3 chapExoCorrec/8982 sacados/8982 EFGH7cm3x1 chapExoCorrec/1976 sacados/1976 chapExoCorrec/4906 sacados/4906 ABC3x−2x14 chapExoCorrec/11779 sacados/11779 4x311 5x1237 chapExoCorrec/375 sacados/375 8x24x14 3x2x6 chapExoCorrec/4893 sacados/4893 5x32x7 7x23x7 4x12x4 12x38x17 chapExoCorrec/11777 sacados/11777 4x33x5 5x12x10 chapExoCorrec/11782 sacados/11782 5x32x7 7x23x7