Solving Systems of Linear Equations in Two Variables- Graphs

But they do result in the same line on a graph. So in summary, the system of equations { (1/2)x-y=3 and x=6+2y results in one line with an infinite number of solutions. The two equations are not linearly independent, but are equivalent and result in the same line on a graph.
  • #1
DS2C

Homework Statement


Solve the system of equations: { (1/2)x-y=3 and x=6+2y

Homework Equations


NA

The Attempt at a Solution


The solution is 3=3, which is an identity, which means that there is an infinite amount of solutions to the system. Here's where my question lies (asked my teacher but she didn't know):
This system of equations results in two graphs, or lines. Their graphs are identical, so on a graph it would look like a single line. But are they two different lines occupying the same space on the plane, or are they the same line? I hope this makes sense. I've attached a screenshot of the problem out of the book for reference.
Screen Shot 2017-12-04 at 2.39.48 PM.png
 

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  • #2
DS2C said:

Homework Statement


Solve the system of equations: { (1/2)x-y=3 and x=6+2y

Homework Equations


NA

The Attempt at a Solution


The solution is 3=3, which is an identity, which means that there is an infinite amount of solutions to the system. Here's where my question lies (asked my teacher but she didn't know):
This system of equations results in two graphs, or lines. Their graphs are identical, so on a graph it would look like a single line. But are they two different lines occupying the same space on the plane, or are they the same line? I hope this makes sense. I've attached a screenshot of the problem out of the book for reference.View attachment 216102

There is only one line. Any point (x,y) that satisfies one of the equations automatically also satisfies the other.
 
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  • #3
The two equations are not linearly independent which is another way of saying that one can be obtained from the other and there is only one straight line. Just multiply the bottom equation by 1/2 and move y to the left and you will see what I mean.
 
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  • #4
So theyre not two lines occupying the same space. They are one line that resulted from two different equations. Or are they the same equation just in different forms since multiplying by 1/2 turns it into the top one?
 
  • #5
DS2C said:
So theyre not two lines occupying the same space. They are one line that resulted from two different equations. Or are they the same equation just in different forms since multiplying by 1/2 turns it into the top one?
It's really only one line. The two equations are equivalent, meaning that any solutions (ordered pairs (x, y)) of one equation are also solutions of the other equation.
 
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  • #6
Ok thank you guys. Cleared that up.
 
  • #7
You are correct, there are an infinite number of solutions. I would say that the two equations are not linearly independent, rather than saying they are the same equation.
 
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Related to Solving Systems of Linear Equations in Two Variables- Graphs

1. What is a system of linear equations in two variables?

A system of linear equations in two variables is a set of two equations that involve two unknowns, typically represented by x and y. The equations are linear, meaning that the highest power of both x and y is 1. The goal is to find the values of x and y that satisfy both equations simultaneously.

2. How do you graph a system of linear equations in two variables?

To graph a system of linear equations in two variables, you first need to rewrite each equation in slope-intercept form (y = mx + b). Then, plot the y-intercept (b) on the y-axis and use the slope (m) to determine a second point on the line. Finally, connect the two points with a straight line. Repeat this process for the second equation and the point where the two lines intersect is the solution to the system.

3. Can a system of linear equations in two variables have more than one solution?

Yes, a system of linear equations in two variables can have infinitely many solutions if the two equations are parallel lines with the same slope. This means that the lines will never intersect and any point on either line will satisfy both equations.

4. How do you know if a system of linear equations in two variables has no solution?

A system of linear equations in two variables has no solution if the two equations are parallel lines with different y-intercepts. This means that the lines will never intersect, and there is no point that satisfies both equations simultaneously.

5. What is the most efficient way to solve a system of linear equations in two variables?

The most efficient way to solve a system of linear equations in two variables is by using the elimination method or the substitution method. In the elimination method, you add or subtract the two equations to eliminate one of the variables, then solve for the remaining variable. In the substitution method, you solve one equation for one of the variables, then substitute it into the second equation to solve for the other variable.

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