Solve each system by graphing. If there is no solution or an...

Solve each system by graphing. If there is no solution or an infinite number of solutions, so state. Use set notation to express solution sets.
$$\left\{\begin{array}{l}2 x-3 y=6 \\ 4 x+3 y=12\end{array}\right.$$

Key Concepts

System of Linear Equations

A system of linear equations consists of two or more equations that share the same variables. The goal is to find values for the variables that satisfy all equations simultaneously. This concept is fundamental in algebra and can represent various real-world relationships.

Graphical Method

The graphical method involves plotting each equation on a coordinate plane and determining their point of intersection. This visual approach is useful for understanding whether the system has a single unique solution (lines intersect at one point), no solution (lines are parallel), or infinitely many solutions (lines overlap entirely).

Intersection of Lines

The point where the graphs of the equations intersect corresponds to the solution of the system. If there is exactly one intersection point, that point represents the unique solution to the system. Multiple or no intersections indicate either infinitely many solutions or no solution, respectively.

Set Notation

Set notation is a formal way to express the solution set of an equation or system. It provides a concise representation of all solutions, such as indicating a unique solution with a set containing that point, or representing the absence of a solution or an infinite number of solutions.

Transcript

00:01 This problem asks us to solve the system of linear equations given by these two equations, and i want us to do it by graphing.

00:08 And so ultimately we're looking at the graph scene where these two lines are going to intersect.

00:15 So my first step, because this is just the easiest form to graph from, y equals mx plus b, this is slope intercept form.

00:25 The reason why this is so great is that it gives us a y intercept.

00:30 So it gives us our point that's on this up and down y -axis, and then it also gives us a slope.

00:36 So that's our rise overrun.

00:38 That's going to allow us to find other points on the line once we've plotted that y intercept.

00:44 So the first thing that we're going to do here is solve for y in order to get it to this form from both of these equations.

00:52 So for this one, we'll have, let's subtract 2x first.

00:57 We'll get negative 3y equal to negative 2x, plus six, and then we'll need to divide by 3, divide by negative 3.

01:07 So let's do that.

01:12 So when we do that over here, you just get y.

01:16 When we do that over here, we're defining each of these terms by negative 3.

01:20 So negative 2x divided by negative 3 is going to give us 2 thirds, x.

01:28 And then a positive 6 divided by negative 3 will give us a negative 2.

01:32 So we have this equation as 2 thirds x minus 2.

01:41 All right.

01:42 Next down here we'll have subtract 4x and divide by 3.

01:51 This time we'll show this a little bit better.

01:53 We're dividing both terms by 3.

01:55 This gives us y equals negative 4 thirds x plus 4...

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