Question 419240: solve the following system of equations
x+3y=4 (1)
x=5-3y (2)
the solution is?? or there is no solution? Found 2 solutions by Theo, duckness73:Answer by Theo(13342) (Show Source):
substitute for x from the second equation into the first equation to get:
x + 3y = 4 becomes:
5 - 3y + 3y = 4
simplify to get:
5 = 4
since this is not true, your equations do not have a common solution and the graph of the lines do not cross.
to graph these equations, solve for y in each equation.
x + 3y = 4 becomes y = (4-x)/3 =
x = 5 - 3y becomes y = (5-x)/3
if we place these lines into the standard slope intercept form, we will see that they have the same slope and a different y intercept and therefore they are parallel to each other meaning they will never intersect.
y = (4-x)/3 becomes y = 4/3 - x/3 which becomes y = -(1/3)*x + 4/3
y = (5-x)/3 becomes y = 5/3 - x/3 which becomes y = -(1/3)*x + 5/3
the slope intercept form of the equations is y = mx + b where m is the slope and b is the y intercept.
you can see that the slope of both these equations is the same and the y intercept is different.
this means the lines are parallel to each other and will never intersect.
You can put this solution on YOUR website! Lets look carefully at these equations:
x + 3y = 4
x = 5 - 3y
If we substitute the expression of x from the second equation into the first, we have:
(5 - 3y) + 3y = 4
Simplifying the equation and collecting like terms we have:
5 - 3y + 3y = 4 or
5 = 4
Since this clearly is a false statement, the system of equations has NO solution.
Another way to look at this problem is to get both equations into slope-intercept form y = mx + b. We have
y = -(1/3)x + (4/3)
y = -(1/3)x + (5/3)
In this case, the slope of each equation is -(1/3). This means that the lines are either parallel or they are the same line. Because the intercept is different ((4/3) in the first equation and (5/3) in the second equation)) the lines are NOT the same line, therefore they must be parallel. Systems of equations representing parallel lines have no solution.
