Question

How do you use substitution to solve the pair of equations: `y=2x-3` and `x^2+y^2=16`?

Answer 1

`(x,y)~~color(green)(""(2.885,2770)) or color(green)(""(-0.485,-3.970))`

Explanation:

Given
[1]`color(white)("XXX")y=2x-3`
[2]`color(white)("XXX")x^2+y^2=16`
Using [1] we know that we can substitute `(2x-3)` for `y` in [2]
[3]`color(white)("XXX")x^2+(2x-3)^2=16`

[4]`color(white)("XXX")rarr x^2+4x^2-12x+9=16`

[5]`color(white)("XXX")rarr5x^2-12x-7=0`

Using the quadratic formula:
[6]`color(white)("XXX")x=(12+-sqrt((-12)^2-4 * 5 * (-7)))/(2 * 5)`

simplifying (and using a calculator)
`color(white)("XXX"){:(x~~2.885," or ",x~~-0.485):}`
using [1] `rarr`
`color(white)("XXX"){:(y~~2.770," or ", y~~-3.970):}`

The image of the given relations (below) indicates that these results are reasonable.