Question

How do you write an equation of a line that passes through points (0,5), (-3,5)?

Answer 1

`y=5`

Explanation:

The equation of a line can be written as

`y - y_0 = m * (x - x_0)`

where `(x_0, y_0)` is any point that lies on the line.

The gradient, `m`, of the line can be found using any two non-identical points that lie on the line.

`m = frac{y_2 - y_1}{x_2 - x_1}`

`= frac{5 - 5}{-3 - 0}`

`= 0`

A gradient of zero indicates that the line is horizontal.

The equation of the line can thus be simplified to

`y = y_0`

In this case, both points have a `y` coordinate of 5. The equation of this line is therefore

`y = 5`.

Answer 2

`color(green)(y=5)`

Explanation:

For this particular example, we could note that the value of the `y` coordinate is a constant: `5`, so the equation is:
`color(white)("XXX")y=5`

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For the general case:
`color(white)("XXX")`Given two points `color(red)(""(x_1,y_1)` and `color(blue)(""(x_2,y_2))`
`color(white)("XXX")`A two-point equation can be written as:
`color(white)("XXXXXX")(y-color(red)(y_1))/(x-color(red)(x_1))=(color(red)(y_1)-color(blue)(y_2))/(color(red)(x_1)-color(blue)(x_2))`

Substituting
`color(white)("XXX")color(red)(""(0,5))` for `color(red)(""(x_1,y_1))` and
`color(white)("XXX")color(blue)(""(-3,5))` for `color(blue)(""(x_2,y_2))`
we have:
`color(white)("XXX")(y-color(red)5)/(x-color(red)0)=(color(red)5-color(blue)(5))/(color(red)(0)-color(blue)(""(-3))`
Simplifying:
`color(white)("XXX")(y-5)/x=0`

`color(white)("XXX")y=5`