Linear Equations in One Variable

Linear equation in one variable is the equation with standard form `color(purple)(mx+b=0)`.

`m` and `b` are some numbers and `x` is a variable.

Examples of linear equations are:

  • `4x+2=0`
  • `-2a-3=0`
  • `3/2 m-5/3=0`

Using equivalence of equations, we can convert some other equations into the standard form:

  • `2x=5` is equivalent to `2x-5=0` (subtract 5 from both sides of equation)
  • `3/2x=5-2/3x` becomes `13/6x-5=0` (move everything to the left and combine like terms)
  • `sqrt(2)x-5=x+2` becomes `(sqrt(2)-1)x-7=0` (move everything to the left and combine like terms)
  • `1/y=2` becomes `-2y+1=0` (multiply both sides by `y` and move everything to the left)

Equation is linear, when it is written in standard form and variable is raised to the first power only.

Following are NOT linear equations:

  • `2x^2+3=0` (variable raised to the second power)
  • `2y-3=3/2y^2` (there is variable, raised to the second power)
  • `1/y+y=2` (if we multiply both sides by `y`, then we will get `1+y^2=2y`, which is not quadratic)

Exercise 1. Determine, whether `2x=-5` is linear and write it in standard form if it is.

Answer: yes; `2x+5=0`.

Exercise 2. Determine, whether `1=2/3 a` is linear and write it in standard form if it is.

Answer: yes; `2/3a-1=0`.

Exercise 3. Determine, whether `x^2=7` is linear and write it in standard form if it is.

Answer: no.

Exercise 4. Determine, whether `1/x+5=x` is linear and write it in standard form if it is.

Answer: no. Multiplying both sides by `x` gives `1+5x=x^2`.

Exercise 5. Determine, whether `3/x=7/3` is linear and write it in standard form.

Answer: yes; `7/3x-3=0`.