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Equations and inequalities: Solving linear equations

The simplest equation to solve is a linear equation. A linear equation is an equation where the power of the variable(letter, e.g. xx) is 1(one). The following are examples of linear equations.

2 x + 2 = 1 2 - x 3 x + 1 = 2 4 3 x - 6 = 7 x + 2 2 x + 2 = 1 2 - x 3 x + 1 = 2 4 3 x - 6 = 7 x + 2
(1)

In this section, we will learn how to find the value of the variable that makes both sides of the linear equation true. For example, what value of xx makes both sides of the very simple equation, x+1=1x+1=1 true.

Since the definition of a linear equation is that if the variable has a highest power of one (1), there is at most one solution or root for the equation.

This section relies on all the methods we have already discussed: multiplying out expressions, grouping terms and factorisation. Make sure that you are comfortable with these methods, before trying out the work in the rest of this chapter.

2 x + 2 = 1 2 x = 1 - 2 ( like terms together ) 2 x = - 1 ( simplified as much as possible ) 2 x + 2 = 1 2 x = 1 - 2 ( like terms together ) 2 x = - 1 ( simplified as much as possible )
(2)

Now we see that 2x=-12x=-1. This means if we divide both sides by 2, we will get:

x = - 1 2 x = - 1 2
(3)

If we substitute x=-12x=-12, back into the original equation, we get:

LHS = 2 x + 2 = 2 ( - 1 2 ) + 2 = - 1 + 2 = 1 and RHS = 1 LHS = 2 x + 2 = 2 ( - 1 2 ) + 2 = - 1 + 2 = 1 and RHS = 1
(4)

That is all that there is to solving linear equations.

Tip: Solving Equations:

When you have found the solution to an equation, substitute the solution into the original equation, to check your answer.

Method: Solving Linear Equations

The general steps to solve linear equations are:

  1. Expand brackets: Expand (Remove) all brackets that are in the equation.
  2. Rearrange: "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 like terms: Group all like terms together and simplify as much as possible.
  4. Factorise: If necessary factorise.
  5. Write solution: Find the solution and write down the answer(s).
  6. Check: Substitute solution into original equation to check answer.

Figure 1
Khan academy video on equations - 1

Exercise 1: Solving Linear Equations

Solve for xx: 4-x=44-x=4

Solution
  1. Step 1. Determine what is given and what is required :

    We are given 4-x=44-x=4 and are required to solve for xx.

  2. Step 2. Determine how to approach the problem :

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

  3. Step 3. Solve the problem :
    4 - x = 4 - x = 4 - 4 ( Rearrange ) - x = 0 ( group like terms ) ∴ x = 0 4 - x = 4 - x = 4 - 4 ( Rearrange ) - x = 0 ( group like terms ) ∴ x = 0
    (5)
  4. Step 4. Check the answer :

    Substitute solution into original equation:

    4 - 0 = 4 4 = 4 4 - 0 = 4 4 = 4
    (6)

    Since both sides are equal, the answer is correct.

  5. Step 5. Write the final answer :

    The solution of 4-x=44-x=4 is x=0x=0.

Exercise 2: Solving Linear Equations

Solve for xx: 4(2x-9)-4x=4-6x4(2x-9)-4x=4-6x

Solution
  1. Step 1. Determine what is given and what is required :

    We are given 4(2x-9)-4x=4-6x4(2x-9)-4x=4-6x and are required to solve for xx.

  2. Step 2. Determine how to approach the problem :

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

  3. Step 3. Solve the problem :
    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 ( Rearrange ) ( 8 x - 4 x + 6 x ) = ( 4 + 36 ) ( group like terms ) 10 x = 40 ( simplify grouped terms ) 10 10 x = 40 10 ( divide both sides by 10 ) x = 4 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 ( Rearrange ) ( 8 x - 4 x + 6 x ) = ( 4 + 36 ) ( group like terms ) 10 x = 40 ( simplify grouped terms ) 10 10 x = 40 10 ( divide both sides by 10 ) x = 4
    (7)
  4. Step 4. Check the answer :

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

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

  5. Step 5. Write the final answer :

    The solution of 4(2x-9)-4x=4-6x4(2x-9)-4x=4-6x is x=4x=4.

Exercise 3: Solving Linear Equations

Solve for xx: 2-x3x+1=22-x3x+1=2

Solution
  1. Step 1. Determine what is given and what is required :

    We are given 2-x3x+1=22-x3x+1=2 and are required to solve for xx.

  2. Step 2. Determine how to approach the problem :

    Since there is a denominator of (3x+13x+1), we can start by multiplying both sides of the equation by (3x+13x+1). But because division by 0 is not permissible, there is a restriction on a value for x. (x≠-13x≠-13)

  3. Step 3. Solve the problem :
    2 - x 3 x + 1 = 2 ( 2 - x ) = 2 ( 3 x + 1 ) 2 - x = 6 x + 2 ( expand brackets ) - x - 6 x = 2 - 2 ( rearrange) - 7 x = 0 ( simplify grouped terms ) x = 0 ÷ ( - 7 ) ∴ x = 0 ( zero divided by any number is 0) 2 - x 3 x + 1 = 2 ( 2 - x ) = 2 ( 3 x + 1 ) 2 - x = 6 x + 2 ( expand brackets ) - x - 6 x = 2 - 2 ( rearrange) - 7 x = 0 ( simplify grouped terms ) x = 0 ÷ ( - 7 ) ∴ x = 0 ( zero divided by any number is 0)
    (9)
  4. Step 4. Check the answer :

    Substitute solution into original equation:

    2 - ( 0 ) 3 ( 0 ) + 1 = 2 2 1 = 2 2 - ( 0 ) 3 ( 0 ) + 1 = 2 2 1 = 2
    (10)

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

  5. Step 5. Write the final answer :

    The solution of 2-x3x+1=22-x3x+1=2 is x=0x=0.

Exercise 4: Solving Linear Equations

Solve for xx: 43x-6=7x+243x-6=7x+2

Solution
  1. Step 1. Determine what is given and what is required :

    We are given 43x-6=7x+243x-6=7x+2 and are required to solve for xx.

  2. Step 2. Determine how to approach the problem :

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

  3. Step 3. Solve the problem :
    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 ( rearrange) - 17 x = 24 ( simplify grouped terms ) - 17 - 17 x = 24 - 17 ( divide both sides by - 17 ) x = - 24 17 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 ( rearrange) - 17 x = 24 ( simplify grouped terms ) - 17 - 17 x = 24 - 17 ( divide both sides by - 17 ) x = - 24 17
    (11)
  4. Step 4. Check the answer :

    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 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
    (12)

    Since both sides are equal to -13417-13417, the answer is correct.

  5. Step 5. Write the final answer :

    The solution of 43x-6=7x+243x-6=7x+2 is,   x=-2417x=-2417.

Solving Linear Equations

  1. Solve for yy: 2y-3=72y-3=7
     
    Click here for the solution
  2. Solve for yy: -3y=0-3y=0
     
    Click here for the solution
  3. Solve for yy: 4y=164y=16
     
    Click here for the solution
  4. Solve for yy: 12y+0=14412y+0=144
     
    Click here for the solution
  5. Solve for yy: 7+5y=627+5y=62
     
    Click here for the solution
  6. Solve for xx: 55=5x+3455=5x+34
     
    Click here for the solution
  7. Solve for xx: 5x=3x+455x=3x+45
     
    Click here for the solution
  8. Solve for xx: 23x-12=6+2x23x-12=6+2x
     
    Click here for the solution
  9. Solve for xx: 12-6x+34x=2x-24-6412-6x+34x=2x-24-64
     
    Click here for the solution
  10. Solve for xx: 6x+3x=4-5(2x-3)6x+3x=4-5(2x-3)
     
    Click here for the solution
  11. Solve for pp: 18-2p=p+918-2p=p+9
     
    Click here for the solution
  12. Solve for pp: 4p=16244p=1624
     
    Click here for the solution
  13. Solve for pp: 41=p241=p2
     
    Click here for the solution
  14. Solve for pp: -(-16-p)=13p-1-(-16-p)=13p-1
     
    Click here for the solution
  15. Solve for pp: 6p-2+2p=-2+4p+86p-2+2p=-2+4p+8
     
    Click here for the solution
  16. Solve for ff: 3f-10=103f-10=10
     
    Click here for the solution
  17. Solve for ff: 3f+16=4f-103f+16=4f-10
     
    Click here for the solution
  18. Solve for ff: 10f+5+0=-2f+-3f+8010f+5+0=-2f+-3f+80
     
    Click here for the solution
  19. Solve for ff: 8(f-4)=5(f-4)8(f-4)=5(f-4)
     
    Click here for the solution
  20. Solve for ff: 6=6(f+7)+5f6=6(f+7)+5f
     
    Click here for the solution

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