problem solving linear equations pdf

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8.E: Solving Linear Equations (Exercises)

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• Page ID 5024

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8.1 - Solve Equations using the Subtraction and Addition Properties of Equality

In the following exercises, determine whether the given number is a solution to the equation.

• x + 16 = 31, x = 15
• w − 8 = 5, w = 3
• −9n = 45, n = 54
• 4a = 72, a = 18

In the following exercises, solve the equation using the Subtraction Property of Equality.

• y + 2 = −6
• a + \(\dfrac{1}{3} = \dfrac{5}{3}\)
• n + 3.6 = 5.1

In the following exercises, solve the equation using the Addition Property of Equality.

• u − 7 = 10
• x − 9 = −4
• c − \(\dfrac{3}{11} = \dfrac{9}{11}\)
• p − 4.8 = 14

In the following exercises, solve the equation.

• n − 12 = 32
• y + 16 = −9
• f + \(\dfrac{2}{3}\) = 4
• d − 3.9 = 8.2
• y + 8 − 15 = −3
• 7x + 10 − 6x + 3 = 5
• 6(n − 1) − 5n = −14
• 8(3p + 5) − 23(p − 1) = 35

In the following exercises, translate each English sentence into an algebraic equation and then solve it.

• The sum of −6 and m is 25.
• Four less than n is 13.

In the following exercises, translate into an algebraic equation and solve.

• Rochelle’s daughter is 11 years old. Her son is 3 years younger. How old is her son?
• Tan weighs 146 pounds. Minh weighs 15 pounds more than Tan. How much does Minh weigh?
• Peter paid $9.75 to go to the movies, which was $46.25 less than he paid to go to a concert. How much did he pay for the concert?
• Elissa earned $152.84 this week, which was $21.65 more than she earned last week. How much did she earn last week?

8.2 - Solve Equations using the Division and Multiplication Properties of Equality

In the following exercises, solve each equation using the Division Property of Equality.

• 13a = −65
• 0.25p = 5.25
• −y = 4

In the following exercises, solve each equation using the Multiplication Property of Equality.

• \(\dfrac{n}{6}\) = 18
• y −10 = 30
• 36 = \(\dfrac{3}{4}\)x
• \(\dfrac{5}{8} u = \dfrac{15}{16}\)

In the following exercises, solve each equation.

• −18m = −72
• \(\dfrac{c}{9}\) = 36
• 0.45x = 6.75
• \(\dfrac{11}{12} = \dfrac{2}{3} y\)
• 5r − 3r + 9r = 35 − 2
• 24x + 8x − 11x = −7−14

8.3 - Solve Equations with Variables and Constants on Both Sides

In the following exercises, solve the equations with constants on both sides.

• 8p + 7 = 47
• 10w − 5 = 65
• 3x + 19 = −47
• 32 = −4 − 9n

In the following exercises, solve the equations with variables on both sides.

• 7y = 6y − 13
• 5a + 21 = 2a
• k = −6k − 35
• 4x − \(\dfrac{3}{8}\) = 3x

In the following exercises, solve the equations with constants and variables on both sides.

• 12x − 9 = 3x + 45
• 5n − 20 = −7n − 80
• 4u + 16 = −19 − u
• \(\dfrac{5}{8} c\) − 4 = \(\dfrac{3}{8} c\) + 4

In the following exercises, solve each linear equation using the general strategy.

• 6(x + 6) = 24
• 9(2p − 5) = 72
• −(s + 4) = 18
• 8 + 3(n − 9) = 17
• 23 − 3(y − 7) = 8
• \(\dfrac{1}{3}\)(6m + 21) = m − 7
• 8(r − 2) = 6(r + 10)
• 5 + 7(2 − 5x) = 2(9x + 1) − (13x − 57)
• 4(3.5y + 0.25) = 365
• 0.25(q − 8) = 0.1(q + 7)

8.4 - Solve Equations with Fraction or Decimal Coefficients

In the following exercises, solve each equation by clearing the fractions.

• \(\dfrac{2}{5} n − \dfrac{1}{10} = \dfrac{7}{10}\)
• \(\dfrac{1}{3} x + \dfrac{1}{5} x = 8\)
• \(\dfrac{3}{4} a − \dfrac{1}{3} = \dfrac{1}{2} a + \dfrac{5}{6}\)
• \(\dfrac{1}{2}\)(k + 3) = \(\dfrac{1}{3}\)(k + 16)

In the following exercises, solve each equation by clearing the decimals.

• 0.8x − 0.3 = 0.7x + 0.2
• 0.36u + 2.55 = 0.41u + 6.8
• 0.6p − 1.9 = 0.78p + 1.7
• 0.10d + 0.05(d − 4) = 2.05

PRACTICE TEST

• \(\dfrac{23}{5}\)
• n − 18 = 31
• 4y − 8 = 16
• −8x − 15 + 9x − 1 = −21
• −15a = 120
• \(\dfrac{2}{3}\)x = 6
• x + 3.8 = 8.2
• 10y = −5y + 60
• 8n + 2 = 6n + 12
• 9m − 2 − 4m + m = 42 − 8
• −5(2x + 1) = 45
• −(d + 9) = 23
• 2(6x + 5) − 8 = −22
• 8(3a + 5) − 7(4a − 3) = 20 − 3a
• \(\dfrac{1}{4} p + \dfrac{1}{3} = \dfrac{1}{2}\)
• 0.1d + 0.25(d + 8) = 4.1
• Translate and solve: The difference of twice x and 4 is 16.
• Samuel paid $25.82 for gas this week, which was $3.47 less than he paid last week. How much did he pay last week?

Contributors and Attributions

Lynn Marecek (Santa Ana College) and MaryAnne Anthony-Smith (Formerly of Santa Ana College). This content is licensed under Creative Commons Attribution License v4.0 "Download for free at http://cnx.org/contents/[email protected] ."

A zeroing feedback gradient-based neural dynamics model for solving dynamic quadratic programming problems with linear equation constraints in finite time

• Original Article
• Published: 27 May 2024

Cite this article

• Shangfeng Du 1 ,
• Dongyang Fu   ORCID: orcid.org/0000-0003-0426-4356 1 ,
• Long Jin 2 ,
• Yang Si 1 &
• Yongze Li 1  

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Gradient-based neural dynamics (GND) models are a classical algorithm for solving optimization problems, but it has non-negligible flaws in solving dynamic problems. In this study, a novel GND model, namely the zeroing feedback gradient-based neural dynamics (ZF-GND) models, is proposed based on the original GND model for tracking down the exact solution of dynamic quadratic programming problem (DQP). Further, a nonlinear projection function is designed to accelerate the convergence of the model. An upper bound on the convergence time of the ZF-GND model is rigorously defined through theoretical analysis. The superior effect of the ZF-GND model in terms of convergence is verified through comparison experiments. Finally, an application of robot motion planning is introduced to verify the practicality of the ZF-GND model.

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Acknowledgements

This research was funded in part by the National Key Research and Development Program of China under Grant No. 2022YFC3103101, Key Special Project for Introduced Talents Team of Southern Marine Science and Engineering Guangdong Laboratory under Contract GML2021GD0809, National Natural Science Foundation of China under Contract 42206187, Key projects of the Guangdong Education Department under Grant No. 2023ZDZX4009.

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Shangfeng Du, Dongyang Fu, Yang Si & Yongze Li

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Du, S., Fu, D., Jin, L. et al. A zeroing feedback gradient-based neural dynamics model for solving dynamic quadratic programming problems with linear equation constraints in finite time. Neural Comput & Applic (2024). https://doi.org/10.1007/s00521-024-09762-3

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Received : 27 July 2023

Accepted : 25 March 2024

Published : 27 May 2024

DOI : https://doi.org/10.1007/s00521-024-09762-3

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