Question

How do you solve `3x - y = 3`, `2x + y = 2` by graphing and classify the system?

Answer 1

See a solution process below:

Explanation:

For each equation we need to solve for two points which solve the equation and plot these points and then draw a line through the points:

Equation 1:

First Point:
For `x = 0`

`(3 * 0) - y = 3`
`0 - y = 3`
`-y = 3`
`color(red)(-1) * -y = color(red)(-1) * 3`
`y = -3` or `(0, -3)`

Second Point:
For `y = 0`

`3x - 0 = 3`
`3x = 3`
`(3x)/color(red)(3) = 3/color(red)(3)`
`x = 1` or `(1, 0)`

graph{(3x-y-3)(x^2+(y+3)^2-0.075)((x-1)^2+y^2-0.075)=0 [-20, 20, -10, 10]}

Equation 2:

First Point:
For `x = 0`

`(2 * 0) + y = 2`
`0 + y = 2`
`y = 2` or `(0, 2)`

Second Point:
For `y = 0`

`2x + 0 = 2`
`2x = 2`
`(2x)/color(red)(2) = 2/color(red)(2)`
`x = 1` or `(1, 0)`

graph{(2x+y-2)(3x-y-3)(x^2+(y-2)^2-0.075)((x-1)^2+y^2-0.075)=0 [-20, 20, -10, 10]}

We can see the points cross at `(1, 0)`

graph{(2x+y-2)(3x-y-3)((x-1)^2+y^2-0.05)=0 [-10, 10, -5, 5]}

A system of two linear equations can have one solution, an infinite number of solutions, or no solution. Systems of equations can be classified by the number of solutions. If a system has at least one solution, it is said to be consistent . If a consistent system has exactly one solution, it is independent.

This system is an independent consistent system.