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Method: solving equations

The general steps to solve equations are:

  1. Expand (Remove) all brackets.
  2. "Move" all terms with the variable to the left hand side of the equation, and all constant terms (the numbers) to the right hand side of the equals sign.Bearing in mind that the sign of the terms will change from ( + ) to ( - ) or vice versa, as they "cross over" the equals sign.
  3. Group all like terms together and simplify as much as possible.
  4. Factorise if necessary.
  5. Find the solution.
  6. Substitute solution into original equation to check answer.

Khan academy video on equations - 1

Solve for x : 4 - x = 4

  1. We are given 4 - x = 4 and are required to solve for x .

  2. Since there are no brackets, we can start with grouping like terms and then simplifying.

  3. 4 - x = 4 - x = 4 - 4 ( move all constant terms ( numbers ) to the RHS ( right hand side ) ) - x = 0 ( group like terms together ) - x = 0 ( simplify grouped terms ) - x = 0 x = 0
  4. Substitute solution into original equation:

    4 - 0 = 4
    4 = 4

    Since both sides are equal, the answer is correct.

  5. The solution of 4 - x = 4 is x = 0 .

Solve for x : 4 ( 2 x - 9 ) - 4 x = 4 - 6 x

  1. We are given 4 ( 2 x - 9 ) - 4 x = 4 - 6 x and are required to solve for x .

  2. We start with expanding the brackets, then grouping like terms and then simplifying.

  3. 4 ( 2 x - 9 ) - 4 x = 4 - 6 x 8 x - 36 - 4 x = 4 - 6 x ( expand the brackets ) 8 x - 4 x + 6 x = 4 + 36 move all terms with x to the LHS and all constant terms to the RHS of the = ( 8 x - 4 x + 6 x ) = ( 4 + 36 ) ( group like terms together ) 10 x = 40 ( simplify grouped terms ) 10 10 x = 40 10 ( divide both sides by 10 ) x = 4
  4. Substitute solution into original equation:

    4 ( 2 ( 4 ) - 9 ) - 4 ( 4 ) = 4 - 6 ( 4 ) 4 ( 8 - 9 ) - 16 = 4 - 24 4 ( - 1 ) - 16 = - 20 - 4 - 16 = - 20 - 20 = - 20

    Since both sides are equal to - 20 , the answer is correct.

  5. The solution of 4 ( 2 x - 9 ) - 4 x = 4 - 6 x is x = 4 .

Solve for x : 2 - x 3 x + 1 = 2

  1. We are given 2 - x 3 x + 1 = 2 and are required to solve for x .

  2. Since there is a denominator of ( 3 x + 1 ), we can start by multiplying both sides of the equation by ( 3 x + 1 ). But because division by 0 is not permissible, there is a restriction on a value for x. ( x - 1 3 )

  3. 2 - x 3 x + 1 = 2 ( 2 - x ) = 2 ( 3 x + 1 ) 2 - x = 6 x + 2 ( remove / expand brackets ) - x - 6 x = 2 - 2 move all terms containing x to the LHS and all constant terms ( numbers ) to the RHS . - 7 x = 0 ( simplify grouped terms ) x = 0 ÷ ( - 7 ) t h e r e f o r e x = 0 zero divided by any number is 0
  4. Substitute solution into original equation:

    2 - ( 0 ) 3 ( 0 ) + 1 = 2 2 1 = 2

    Since both sides are equal to 2, the answer is correct.

  5. The solution of 2 - x 3 x + 1 = 2 is x = 0 .

Solve for x : 4 3 x - 6 = 7 x + 2

  1. We are given 4 3 x - 6 = 7 x + 2 and are required to solve for x .

  2. We start with multiplying each of the terms in the equation by 3, then grouping like terms and then simplifying.

  3. 4 3 x - 6 = 7 x + 2 4 x - 18 = 21 x + 6 ( each term is multiplied by 3 ) 4 x - 21 x = 6 + 18 ( move all terms with x to the LHS and all constant terms to the RHS of the = ) - 17 x = 24 ( simplify grouped terms ) - 17 - 17 x = 24 - 17 ( divide both sides by - 17 ) x = - 24 17
  4. Substitute solution into original equation:

    4 3 × - 24 17 - 6 = 7 × - 24 17 + 2 4 × ( - 8 ) ( 17 ) - 6 = 7 × ( - 24 ) 17 + 2 ( - 32 ) 17 - 6 = - 168 17 + 2 - 32 - 102 17 = ( - 168 ) + 34 17 - 134 17 = - 134 17

    Since both sides are equal to - 134 17 , the answer is correct.

  5. The solution of 4 3 x - 6 = 7 x + 2 is,    x = - 24 17 .

Solving linear equations

  1. Solve for y : 2 y - 3 = 7
  2. Solve for w : - 3 w = 0
  3. Solve for z : 4 z = 16
  4. Solve for t : 12 t + 0 = 144
  5. Solve for x : 7 + 5 x = 62
  6. Solve for y : 55 = 5 y + 3 4
  7. Solve for z : 5 z = 3 z + 45
  8. Solve for a : 23 a - 12 = 6 + 2 a
  9. Solve for b : 12 - 6 b + 34 b = 2 b - 24 - 64
  10. Solve for c : 6 c + 3 c = 4 - 5 ( 2 c - 3 )
  11. Solve for p : 18 - 2 p = p + 9
  12. Solve for q : 4 q = 16 24
  13. Solve for q : 4 1 = q 2
  14. Solve for r : - ( - 16 - r ) = 13 r - 1
  15. Solve for d : 6 d - 2 + 2 d = - 2 + 4 d + 8
  16. Solve for f : 3 f - 10 = 10
  17. Solve for v : 3 v + 16 = 4 v - 10
  18. Solve for k : 10 k + 5 + 0 = - 2 k + - 3 k + 80
  19. Solve for j : 8 ( j - 4 ) = 5 ( j - 4 )
  20. Solve for m : 6 = 6 ( m + 7 ) + 5 m

Questions & Answers

what is demand
Joseph Reply
what is mean by unitary elastic demand
Bangniyel Reply
demand is said to be unitary elastic when the percentage change in the demand is equal to the percentage change in the price
George
what is the principle of equi-marginal utility
Reliance Reply
what is Economics and it important
Anita Reply
what is production
Anita
what is Economic
Anita Reply
what is the meaning of Economic
Anita
economics is a science which studies human behaviour as a relationship between ends and scarce means which have alternative uses
Boso
I don't know.
natchanan
u don't
Ssali
@ Boso thanks for the definition ✌
Raewyn
boso u r too much u try
Liambee
ya nyc
Ssali
Thanks kk
Anita
pls can I ask more questions
Anita
what is production
Anita
production is creation of goods and services
George
what is macroeconomics and microeconomics
Anita
macroeconomics deals with larger economic units such as GDP,GNP,employment while microeconomics deals with smaller economic units such firm and household
George
Thanks
Anita
Explain the ff Scarcity Ends Demand Supply Choice Scale of preference
Anita
macroeconomics deals with larger economic units such as GDP,GNP,employment while microeconomics deals with smaller economic units such firm and household
George
Gross Domestic product...it represent the total value of the products produced within the country including foreign industries
George Reply
what is products
Anita
what are the favourable demand
Odia Reply
list of climate that affect demands
Odia
What is two major forms of international trade?
Musa Reply
What is Economics and why it is important
Abdul Reply
De ans,Economics is the study of women behavior as a relationship between end and scared mean which have alternative uses.
Anita
What is Inflation
Abdul Reply
More money = more consumers, more consumers = lessen the product, less product = high price, high price = inflation
TuroN
is it definition
Odia
why price and quantity increase
Otuu Reply
condition under which price and quantity will be increased at the same time
Otuu
factors that hinders mobility of labour
Dennis Reply
what is scarcity
Adams Reply
why are people still unemployed in this economy we live in
Joyce Reply
why are people still unemployed in this economy
Joyce
Because of high technology
Innocent
Due to low government expenditure, the state does not have money to invest in new projects that can result in more job opportunities.
Young
good afternoon all, what are the examples of development projects Pls?
James
I 'm not together with him, when we say government expenditure mean there is recession for her economic country. That why many people still unemployed in this economy we live in, it mainly depend for two phases like Natural hazards and human activities.
Innocent
For the first answer I was responded we are concerns to intensive technology in the whole world it lead to reduce of many workers.
Innocent
wat does the abbreviation ppc mean?
Tumiso
I would like to give you ideas so that you get well how can give answer to this question dears. Attempt to know these words and what these means; Deficit, Surplus and Balanced budget.
Innocent
ppc mean pay -per-click known as search engine marketing
Innocent
i found it ppc means production possibility curve
Tumiso
in economic is that true according to your answer, however about of Marketing is that Pay Per Click.
Innocent
how is inflation, output gap relate to monetary policy
Mary Reply
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Source:  OpenStax, Linear equations. OpenStax CNX. Jun 15, 2015 Download for free at https://legacy.cnx.org/content/col11828/1.1
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