Question 86726
The form of the equation that you need for this problem is called the slope-intercept
form. It is given by the equation:
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y = mx + b
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where m is the slope of the graph of the equation and b is the y-intercept, that is, the
value on the y-axis where the graph crosses.
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The first part of the problem gives you the equation:
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2x - 3y = 6 
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and asks you to find the slope and the y-intercept of the graph.  The easiest way to do that
is to rearrange the equation into the slope intercept form. Once you do that you know that
the slope (m) is the multiplier of the x and the constant is the y-intercept
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So we need to solve the given equation for y. The first step is to isolate the y term on
one side of the equation and have everything else on the other side.  So let's begin by 
subtracting 2x from both sides of the equation.  When you do that the equation becomes:
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-3y = -2x + 6
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Next solve for y by dividing all terms on both sides of the equation by -3, the multiplier
of the y term.  When you do that division the equation becomes:
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-3y/3 = -2x/-3 + 6/-3
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and this simplifies to:
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y = (2/3)*x - 2
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Now by comparing this equation with the slope-intercept form you can tell that the slope
(the multiplier of x represented by m) is (2/3) and the y-intercept (the constant 
represented by b) is -2. Those are the answers to the first part of this problem.
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The remainder of the problem can be done just by substituting values into the slope-intercept
equation form.  
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The first of the three givens tells you that the slope is 5 and the y-intercept is (0, -2).
The intercept point on the y-axis is, therefore, -2.  The slope intercept equation form
becomes:
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y = 5x - 2
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You can graph this by marking the point -2 on the y-axis. You know that the graph goes
through this point. Put your pencil on that point. Move in horizontally to the right
1 unit. Stop. Then move it vertically upward 5 units. Stop and mark that point. This is
a second point on the graph.  Now you can run a straight line through the two points and 
you have the graph.  Another way you can find points is to just assume values for x, 
plug them into the equation, and solve for y.  The point (x, y) will then be on the graph. 
For example, assume x = 2. Then the equation for y tells you that y = 5(2)-2 = 10 - 2 = 8.
So the point (2,8) is on the graph.  When you get done, the graph should look like:
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{{{graph(300,300,-9, 9,-20,20,5x -2)}}}
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You can do the remaining two problems the same way.  For the second one you are given that
the slope is -2 and the intercept is (0,4).  That means the slope intercept equation is:
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y = -2x + 4
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Since the graph crosses the y axis at +4 you can mark that point on the y-axis. Start your 
pencil on that location. Go horizontally to the right 1 unit and at that location go
down -2 units because the slope is -2.  (You should be at (1,2)). That is a second 
point on the graph.  Since you now have two points on the graph you can draw a line through
these points and that is the graph you need.  Again you can also find additional points by 
assuming values for x and solving for corresponding values of y.  Then plot the (x,y) points. 
When you get done the graph should look like:
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{{{graph(300,300,-9, 9,-20,20,-2x +4)}}}
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The final problem gives you a slope of -3/4 and a y-intercept of 8. The equation for this 
is:
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y = -(3/4)x + 8
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Mark +8 on the y-axis. Then go horizontally to the right from that point 1 unit followed
by going vertically down 3/4 of a unit. Mark that point.  (or you can go horizontally
to the right 4 units from the point on the y-axis and then go vertically down 3 units).  
You can also substitute values for x into the equation and find corresponding values
of y to give you more points on the graph.  When you get done, your graph should look like:
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{{{graph(300,300,-20, 20,-20,20,-0.75x +8)}}}
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Hope this helps you to understand the slope-intercept form a little better.