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Question 617761: Without graphing answer the following questions about the system:
4x+y=7
32x-56=-8y
Does the system have one solution, no solution or an infinite number of solutions? Is the system incosistent, are the equations dependent, or neither?
What does the eqation look like? Does it have one line, parallel lines or intersecting lines?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! your equations are:
4x+y=7
32x-56=-8y
solve for y in both equations to put them both in the slope intercept form for linear equations of y = mx + b
m is the slope and b is the y intercepts.
the first equation is:
4x + y = 7
subtract 4x from both sides of the equation to getr:
y = -4x + 7
the slope is -4
the y intercept is 7
the second equation is:
32x - 56 = -8y
add 8y to both sides of the equation to get:
8y + 32x - 56 = 0
subtract 32x and add 56 from both sides of the equation to get:
8y = -32x + 56
divide both sides of the equation by 8 to get:
y = -32x/8 + 56/8
simplify this to get:
y = -4x + 7
the equations are identical.
they have the same slope and the same y intercept
if you graph them they will look like the same line.
the system of equations has an infinite number of solutions because the lines are identical and they coincide completely with each other.
in the standard form for the equation of a straight line, if the equations are exact multiples of each other, then they are identical.
in the standard form, you get:
4x + y = 7 (already in standard form)
32x + 8y = 56 (converted to standard form_)
32x / 4x = 8
8y / y = 8
56 / 7 = 8
since each term in the equations have the same ratio, then the equations are in proportion and are therefore identical to each other.
when you place them in the slope intercept form of the equation, they become identical meaning they are the same line.
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