# 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`.