Equations

What number makes the following statement below true?

2x+3=9

Substituting various values for x, we find that only x=3 works, since 2(3)+3=6+3=9

We say that x=3 is the solution of the equation 2x+3=9.

Solving an equation is the same as making x the subject of formula.

  • 2x+3=9
  • 2x=9-3
  • 2x=6
  • x=6/2
  • x=3

Let’s look at some examples.

Example

Solve the following equations:

i) 7x-4=10

ii) 15x-8=5x+2

iii) \frac{x+1}{3}=-2

i) 7x-4=10

7x=10+4

7x=14

x=\frac{14}{7}

x=2

ii) 15x-8=5x+2

We isolate all the x’s on the right hand side.

15x-5x=2+8

10x=10

x=\frac{10}{10}

x=1

iii) \frac{x+1}{3}=-2

Multiply both sides by 3

x+1=-2(3)

x+1=-6

x=-6-1

x=-7


Example

Find the value of x if:

\frac{2x-1}{5} + \frac{3x+2}{3} = \frac{1}{2}

We shall eliminate all the numbers in the denominators by multiplying every term by 30.

30(\frac{2x-1}{5})+30(\frac{3x+2}{3})=\frac{1}{2}(30)

6(2x-1)+10(3x+2)=15

12x-6+30x+20=15

42x+14=15

42x=15-14

42x=1

x=\frac{1}{42}


Example

Find the value of x in the equation:

\frac{5x-1}{6}=\frac{1-x}{4}

If we multiply both sides by 12, we obtain:

12(\frac{5x-1}{6})=12(\frac{1-x}{4}

2(5x-1)=3(1-x)

10x-2=3-3x

10+3x=3+2

13x=5

x=\frac{5}{13}