Sketch the following lines on the same diagram: 2 x-3 y=6, 4 x-6 y=18, x-(3)/(2) y=3 Hence c

Sketch the following lines on the same diagram: 2 x-3 y=6, ...

Key Concepts

Dependent Systems

Dependent systems occur when two or more equations represent the same line (i.e., they are multiples of each other). In such cases, the system has infinitely many solutions since every point on the line satisfies all the equations. Recognizing when an equation can be transformed into another is key to identifying dependent systems.

Parallel Lines

Parallel lines have identical slopes but different y-intercepts, meaning they never intersect. In the context of systems of linear equations, if two equations have the same slope and different intercepts, the system has no solution and is considered inconsistent.

Consistent and Inconsistent Systems

A system of equations is consistent if there is at least one solution, and inconsistent if there is no solution. When the equations represent parallel lines or lines that intersect at a common point, the nature of their interaction determines whether the system has a unique solution, infinitely many solutions (if they are the same line), or no solution at all (if they are parallel).

Graphing of Linear Equations

Graphing linear equations involves plotting lines on the coordinate plane based on their algebraic form. It allows for a visual interpretation of the solutions of the system. The intercepts and slopes are extracted to accurately plot each line, which helps in understanding how the lines interact—whether they intersect, overlap, or remain parallel.

Slope and y-intercept

The slope and y-intercept form of a linear equation reveals the angle and position of a line in the coordinate plane. The slope indicates the direction and steepness, while the y-intercept shows where the line crosses the y-axis. This information is critical when comparing lines to determine if they are parallel or coincident.

Transcript

00:01 Given 2x minus 3y equals 6 and x minus 3 halves y equals 3, we're going to find the system, the solution to the system of linear equations.

00:11 So i'm actually going to rewrite this over here, x minus 3 halves y equals 3.

00:18 And i'm going to solve both of these equations for why because i like to compare them in slope intercept form.

00:24 So over here, i'm going to subtract 2x from both sides.

00:27 It gives me negative 3y equals 6 minus 2x.

00:31 I'm going to divide every term by negative three.

00:36 We get y equals two thirds x minus two.

00:41 Over here, i'm going to solve for y by subtracting x from both sides.

00:46 Negative three halves y equals negative x plus three.

00:51 I'm going to divide by negative three halves, which is really just multiplying both sides by negative two thirds because dividing fractions is the same as multiplying by the reciprocal.

01:05 I get y equals, we've got to distribute this negative two thirds to both pieces.

01:10 So i get two thirds x minus six over three, which is y equals two thirds x minus two.

01:19 And now if we look at these equations, one right on top of the other like this, we can see that they are exactly the same.

01:26 I've got one y on both sides.

01:28 My slope is the same at two thirds and my inters.

01:32 My y intercept is the same at negative two.

01:34 So because they're the same line, these are, this has infinitely many solutions.

01:47 If i were to graph this, it crosses down here at negative two and my slope is two thirds up to one two over one, two, three, i would get a point here, down to and over one, two, three.

02:07 I would get a point here.

02:10 So that's the top line.

02:12 And then we'll draw the next line in a different color.

02:14 We'll use pink here.

02:15 And if i would draw this line, i would get the exact same points.

02:18 It would lay right on top of it.

02:19 It's the exact same line.

02:25 Now we're going to look at 4x minus 6y equals 18 and x minus 3 over 2y equals 3...