SOLUTION: Hi my teacher wants us to determin what each linear graph is, But I have always had trouble with linear equations I was wondering if you could just send me an example of the three
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Question 250122: Hi my teacher wants us to determin what each linear graph is, But I have always had trouble with linear equations I was wondering if you could just send me an example of the three different graphs and an explantion for each (independant, dependent, and inconsistent). So I can try to do it myself. thank you so much!!
Found 2 solutions by stanbon, Theo:
Answer by stanbon(75887) (Show Source): You can put this solution on YOUR website!
Hi my teacher wants us to determin what each linear graph is, But I have always had trouble with linear equations I was wondering if you could just send me an example of the three different graphs and an explantion for each (independant, dependent, and inconsistent). So I can try to do it myself. thank you so much!!
----------------
Independent: the slopes are different
y = 3x+6
y = x +6
------------------
Dependent: one equation is a multiple of the other
y = 3x+6
2y = 6x+12
-------------------
Inconsistent: the slopes are the same and the intercepts are different.
y = 3x+6
y = 3x+8
==================
Cheers,
Stan H.
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
linear equations in two dimensions are equations of lines.
they have the form ax + by = c where a is the coefficient of the x term and b is the coefficient of the y term and c is a constant.
the slope intercept form of these equations is y = m*x + b where m is the slope and b is the y-intercept (value of y when x = 0).
when you graph these equations, you use the slope-intercept form.
to convert from the standard form of the equation to the slope-intercept form of the equation, you solve for y.
no solution occurs when the lines do not intercept. this means that they are parallel to each other.
here's a graph of 2 linear equations where there is no solution.
the equations for these lines are:
y = x+3
y = x-3
not the slopes are equal.
since the lines are parallel, they will never intersect.
independent solution occurs when the lines do intersect. this means that there is one unique solution to the problem.
here's a graph of 2 linear equations where the lines intersect.
the equations for these lines are:
y = x+3
y = -x-3
note that the slopes are not equal.
these lines are not parallel and they are not identical so they will always intersect.
dependent solutions occur when the lines are identical.
there are multiple solutions to these equations.
each equation is usually identical to, or an exact multiple of, the other.
these lines have multiple solutions because they are the same line.
an example of 2 such equations in standard form would be:
5x + 3y = 7
10x + 6y = 14
solve for y in both equations and you will get an identical equation.
the first equation becomes y = -5x/3 + 7/3
the second equation becomes y = -10x/6 + 14/6
since 10/6 is equivalent to 5/3 and 14/6 is equivalent to 7/3, the second equation simplifies to y = -5x/3 + 7/3
these lines will have multiple solutions because they are the same line so the lines will intersect at all points.
here's a graph of the identical lines.
it looks like one line but it is really 2 identical lines superimposed on each other.
slope intercept form of the no solution equations are:
y = x-3
y = x+3
standard form of these equations is:
-x + y = -3
-x + y = 3
subtract second equation from first equation and you get:
0 + 0 = -6 which becomes 0 = -6 which is NOT true therefore these is no solution.
slope intercept form of the independent solution equations is:
y = x+3
y = -x-3
standard form of these equations is:
-x + y = 3
x + y = -3
subtract second equation from first equation to get:
-2x = 6
solve for x to get x = -3
solve for y to get y = 0
there is a unique solution at (x,y) = (-3,0) because the lines intersect at that point and 2 lines in the same plane can intersect in one and only one point.
standard form of the dependent solution equations is:
5x + 3y = 7
10x + 6y = 14
multiply first equation by 2 to get:
10x + 6y = 14
10x + 6y = 14
subtract second equation from first equation to get:
0 + 0 = 0 which becomes 0 = 0
the variable dropped out and the equation IS true, so there are multiple solutions to this problems.
any value of x will lead to a value of y that automatically applies to both equations because these equations are identical.
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