which graph represents a system of linear equations that has multiple common coordinate pairs.
A system of linear equations has multiple common coordinate pairs when the equations are dependent, meaning that one equation is a multiple of the other. In terms of graph representation, this means that the two equations will have the same line graph.
To illustrate this, let's consider the following system of linear equations:
Equation 1: y = 2x + 3
Equation 2: 2y = 4x + 6
Both equations have the same slope-intercept form, which implies that the lines they represent will be identical. We can rearrange Equation 2 to match the form of Equation 1:
Equation 2 (rearranged): y = 2x + 3
As you can see, the equations are equivalent, so their graphs will be the same line. In other words, any coordinate pair that satisfies one equation will also satisfy the other equation.
To determine which graph represents a system of linear equations that has multiple common coordinate pairs, you will need to understand the concept of a system of linear equations and how it is represented graphically.
A system of linear equations consists of two or more linear equations with the same variables. The common coordinate pairs, also known as solutions or intersection points, are the points that satisfy all the equations in the system.
To graph a linear equation, we represent it as a straight line on a coordinate plane. Each line represents an equation in the system, and the common coordinate pairs are the points where these lines intersect.
Now, to find a system of linear equations that has multiple common coordinate pairs, you need to look for graphs that intersect at more than one point. Let's consider an example:
System of linear equations:
Equation 1: y = 2x + 1
Equation 2: y = -3x + 4
To graph these equations, you can choose any range of x-values and substitute them into each equation to find the corresponding y-values. Once you have a few coordinate pairs, plot them on a graph, and draw a straight line passing through these points.
For the given example equations:
- Plot the points (0,1) and (1,3) for Equation 1 (y = 2x + 1).
- Plot the points (0,4) and (1,1) for Equation 2 (y = -3x + 4).
Now, draw the lines passing through these points on the graph. In this case, you will notice that the lines intersect at the point (1,3). This means that (1,3) is a common coordinate pair, or a solution, for both equations.
To check for multiple common coordinate pairs, you can choose another set of points and repeat the process. For example, you can take the points (2,5) and (3,7). By plotting these points and drawing the lines, you will find that the lines intersect again at the point (2,5). Therefore, (2,5) is another common coordinate pair for this system of linear equations.
In conclusion, the graph that represents a system of linear equations with multiple common coordinate pairs will show multiple points of intersection between the lines representing the equations.