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Solving linear equations

Module by: Free High School Science Texts Project. E-mail the author

Strategy for Solving Equations

This chapter is all about solving different types of equations for one or two variables. In general, we want to get the unknown variable alone on the left hand side of the equation with all the constants on the right hand side of the equation. For example, in the equation x-1=0x-1=0

 
, we want to be able to write the equation as x=1x=1.

As we saw in review of past work (section on rearranging equations), an equation is like a set of weighing scales that must always be balanced. When we solve equations, we need to keep in mind that what is done to one side must be done to the other.

Method: Rearranging Equations

You can add, subtract, multiply or divide both sides of an equation by any number you want, as long as you always do it to both sides.

For example, in the equation x+5-1=-6x+5-1=-6

 
, we want to get xx alone on the left hand side of the equation. This means we need to subtract 5 and add 1 on the left hand side. However, because we need to keep the equation balanced, we also need to subtract 5 and add 1 on the right hand side.

x + 5 - 1 = - 6 x + 5 - 5 - 1 + 1 = - 6 - 5 + 1 x + 0 + 0 = - 11 + 1 x = - 10 x + 5 - 1 = - 6 x + 5 - 5 - 1 + 1 = - 6 - 5 + 1 x + 0 + 0 = - 11 + 1 x = - 10
(1)

In another example, 23x=823x=8, we must divide by 2 and multiply by 3 on the left hand side in order to get xx alone. However, in order to keep the equation balanced, we must also divide by 2 and multiply by 3 on the right hand side.

2 3 x = 8 2 3 x ÷ 2 × 3 = 8 ÷ 2 × 3 2 2 × 3 3 × x = 8 × 3 2 1 × 1 × x = 12 x = 12 2 3 x = 8 2 3 x ÷ 2 × 3 = 8 ÷ 2 × 3 2 2 × 3 3 × x = 8 × 3 2 1 × 1 × x = 12 x = 12
(2)

These are the basic rules to apply when simplifying any equation. In most cases, these rules have to be applied more than once, before we have the unknown variable on the left hand side of the equation.

Tip:

The following must also be kept in mind:
  1. Division by 0 is undefined.
  2. If xy=0xy=0, then x=0x=0 and y0y0, because division by 0 is undefined.

We are now ready to solve some equations!

Investigation : Strategy for Solving Equations

In the following, identify what is wrong.

4 x - 8 = 3 ( x - 2 ) 4 ( x - 2 ) = 3 ( x - 2 ) 4 ( x - 2 ) ( x - 2 ) = 3 ( x - 2 ) ( x - 2 ) 4 = 3 4 x - 8 = 3 ( x - 2 ) 4 ( x - 2 ) = 3 ( x - 2 ) 4 ( x - 2 ) ( x - 2 ) = 3 ( x - 2 ) ( x - 2 ) 4 = 3
(3)

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
(4)

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 a possible ) 2 x + 2 = 1 2 x = 1 - 2 ( like terms together ) 2 x = - 1 ( simplified as much a possible )
(5)

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
(6)

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

L H S = 2 x + 2 = 2 ( - 1 2 ) + 2 = - 1 + 2 = 1 a n d R H S = 1 L H S = 2 x + 2 = 2 ( - 1 2 ) + 2 = - 1 + 2 = 1 a n d R H S = 1
(7)

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 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.

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 grouping like terms and then simplifying.

  3. Step 3. Solve the problem :
    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 - 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
    (8)
  4. Step 4. Check the answer :

    Substitute solution into original equation:

    4 - 0 = 4 4 - 0 = 4
    (9)
    4 = 4 4 = 4
    (10)

    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 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 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 ( 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
    (11)
  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
    (12)

    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 ( 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 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
    (13)
  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
    (14)

    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 ( 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 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
    (15)
  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
    (16)

    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 ww: -3w=0-3w=0
     
    Click here for the solution
  3. Solve for zz: 4z=164z=16
     
    Click here for the solution
  4. Solve for tt: 12t+0=14412t+0=144
     
    Click here for the solution
  5. Solve for xx: 7+5x=627+5x=62
     
    Click here for the solution
  6. Solve for yy: 55=5y+3455=5y+34
     
    Click here for the solution
  7. Solve for zz: 5z=3z+455z=3z+45
     
    Click here for the solution
  8. Solve for aa: 23a-12=6+2a23a-12=6+2a
     
    Click here for the solution
  9. Solve for bb: 12-6b+34b=2b-24-6412-6b+34b=2b-24-64
     
    Click here for the solution
  10. Solve for cc: 6c+3c=4-5(2c-3)6c+3c=4-5(2c-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 qq: 4q=16244q=1624
     
    Click here for the solution
  13. Solve for qq: 41=q241=q2
     
    Click here for the solution
  14. Solve for rr: -(-16-r)=13r-1-(-16-r)=13r-1
     
    Click here for the solution
  15. Solve for dd: 6d-2+2d=-2+4d+86d-2+2d=-2+4d+8
     
    Click here for the solution
  16. Solve for ff: 3f-10=103f-10=10
     
    Click here for the solution
  17. Solve for vv: 3v+16=4v-103v+16=4v-10
     
    Click here for the solution
  18. Solve for kk: 10k+5+0=-2k+-3k+8010k+5+0=-2k+-3k+80
     
    Click here for the solution
  19. Solve for jj: 8(j-4)=5(j-4)8(j-4)=5(j-4)
     
    Click here for the solution
  20. Solve for mm: 6=6(m+7)+5m6=6(m+7)+5m
     
    Click here for the solution

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