Welcome to a lesson on writing linear equations
in sloped-intercept form.
We're asked to give the equation of the line
in slope-intercept form from the given information.
The linear equation is in slope-intercept form,
if it's in the form y equals m x plus b,
where m, the coefficient of x is equal to the slope,
and b, the constant term,
is equal to the vertical intercept.
As an ordered pair, the vertical intercept would be 0,b,
so in order to determine the equation of a line
in slope-intercept form,
we need to find the slope and the vertical intercept,
so looking at A,
we're trying to find the equation of a line
that has a vertical intercept of 0,2,
and a slope of negative nine,
so notice, in this case,
we're given all the information we need,
because the intercept is the point 0,2,
b equals two, and because the slope equals negative nine,
m equals negative nine,
which means the equation of the line in slope-intercept form
is y equals negative nine x plus two.
In B, we want to find the equation of a line
passing through the point 2,3,
which notice is not the vertical intercept
with slope negative five,
so we are given the slope,
so we know that m is equal to negative five,
but again, the point 2,3 is not the vertical intercept,
so we don't know b,
but we do know our equation so far
would have to be in the form
y equals negative five x plus b.
Now, to determine the value of b,
we'll do this algebraically.
Since the point 2,3 is on this line,
that means that this ordered pair
must satisfy this equation,
so if we substitute two for x and three for y,
that will leave us with one unknown, b,
which we can then solve for,
so we'll take the point 2,3,
in this equation here, and perform substitution,
so we'll substitute three for y,
that would be three equals, and two for x,
so negative five times two plus b,
now we can solve for b,
so we have three equals, this'd be negative ten plus b.
Solving for b, we add ten to both sides.
Simplifying, we have 13 equals this'd be zero,
so 13 equals b.
So, now that we know that b equals 13,
we now know the equation of our line would be
y equals negative five x plus 13.
Let's verify this one on the graphic calculator.
To do this, we'll graph the linear equation
that we just found, and verify the point 2,3
would be on the line.
So, we'll press y equals and run our linear equation,
which is negative five x plus 13.
To make sure we have the correct window,
or the standard window, we can press zoom six,
and here's the graph of our line,
and there're several ways to verify the point 2,3
is on this line.
Let's press second, trace,
and notice how value is highlighted,
so we'll press enter,
and now we'll want to enter the x value of two,
our x coordinate,
and if we press enter,
notice how it tells us the corresponding y coordinate
is three, and, threfore, the point 2,3 is on the line,
and we can see the cursor on that point right here,
verifying our equation, it is correct.
Now, for c, you want to find the equation
of a line that contains these two points.
Notice, neither of these points are the vertical intercept,
and we're also not given the slope,
so our first step is to find the slope.
To do this, we'll be using our slope formula given here,
so let's go ahead and label the ordered pairs.
Let's call this x sub one, y sub one,
and x sub two, y sub two,
which means the slope of our line
passing through these two points
would be equal to this ratio here,
where we'd have y sub two minus y sub one,
that's 16 minus six divided by x sub two minus x sub one,
which would be four minus two.
Simplifying, 16 minus six is ten.
Four minus two is two.
Ten divided by two equals five.
So, now that we know the slope,
we know the equation of our line
must be in the form of y equals five x plus b,
and just like last time,
to find the vertical intercept, b,
we'll select one of the two points,
it doesn't matter which one,
perform substitution and then solve for b.
Let's go ahead and use this first point here,
so using the point 2,6, we'll substitute two for x,
six for y, and solve for b,
so we would have six equals five times two plus b.
Simplifying, we have six equals ten plus b,
subtracting ten on both sides and simplifying.
We have six minus ten, that's negative four equals b,
so now that we know that the slope is five,
and the vertical intercept is negative four,
we have all the information we need.
M equals five, b equals negative four,
and, therefore, our equation will be five x minus four.
This was quite a bit of work,
so, again, let's verify this on the graphing calculator.
Let's press y equals, let's enter our equation,
so we'll clear the old equation,
and we'll enter five x minus four.
Let's go ahead and press graph,
so here's the graph of our equation.
Notice how the second point here
is not going to be on the screen,
so this might not work,
but let's go ahead and try it.
Let's first check the first point,
so we'll press second, trace for the calculation menu,
press enter for the value feature,
enter the x value of two, enter.
Notice how our line does contain the point 2,6,
and now let's try the second point.
So again, we'll press second, trace, enter for value,
and let's try pressing four.
We might get an error, because, notice,
the y coordinate of 16 would not be on the vertical axis.
Right now, the vertical axis only goes
from negative ten to positive ten,
so if we press enter, it actually did give it to us,
but it didn't show on the screen,
so this does verify our linear equation is correct.
I hope you found this helpful.