Question

How do you solve the system of equations by graphing and then classify the system as consistent or inconsistent `3x - 2y = 10` and `5x + 2y = 6`?

Answer 1

The lines intersect at a single point, therefore the system of equations is consistent.

Explanation:

Equation 1: `3x-2y=10`

Equation 2 : `5x+2y=6`

Both equations are in the standard form for a linear equation. This form makes it easy to determine the x- and y-intercepts. We can use those two points to graph each equation.

X-intercept: value of `x` when `y=0`

Substitute `0` for `y` and solve for `x`.

Y-intercept: value of `y` when `x=0`

Substitute `0` for `x` and solve for `y`.

Equation 1

`3x-2y=10`

X-intercept: Substitute `0` for `y` and solve for `x`.

`3x-2(0)=10`

`3x=10`

Divide both sides by `3`.

`x=10/3` or `~~3.333`

x-intercept: `(10/3,0)` or `(~~3.333,0)` Plot this point.

Y-intercept: Substitute `0` for `x` and solve for `y`.

`3(0)-2y=10`

`-2y=10`

Divide both sides by `-2`.

`y=10/(-2)`

`y=-5`

y-intercept: `(0,-5)` Plot this point.

Draw a straight line through the two points. This is the graph for Equation 1.

Equation 2

`5x+2y=6`

X-intercept: Substitute `0` for `y` and solve for `x`.

`5x+2(0)=6`

`5x=6`

Divide both sides by `5`.

`x=6/5` or `1.2`

x-intercept: `(6/5,0)` or `(1.2,0)` Plot this point.

Y-intercept: Substitute `0` for `x` and solve for `y`.

`5(0) + 2y=6`

`2y=6`

Divide both sides by `2`.

`y=6/2`

`y=3`

y-intercept: `(0,3)` Plot this point.

Draw a line between the two points. This is the graph of Equation 2.

The lines intersect at a single point, therefore the system of equations is consistent.

graph{(3x-2y-10)(5x+2y-6)=0 [-10, 10, -5, 5]}