Algebra I
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CA Algebra I: Number Properties and Absolute Value
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CA Algebra I: Simplifying Expressions
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CA Algebra I: Simple Logical Arguments
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CA Algebra I: Graphing Inequalities
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CA Algebra I: Slope and Y-intercept
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CA Algebra I: Systems of Inequalities
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CA Algebra I: Simplifying Expressions
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CA Algebra I: Factoring Quadratics
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CA Algebra I: Completing the Square
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CA Algebra I: Quadratic Equation
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CA Algebra I: Quadratic Roots
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CA Algebra I: Rational Expressions 1
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CA Algebra I: Rational Expressions 2
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CA Algebra I: Word Problems
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CA Algebra I: More Word Problems
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CA Algebra I: Functions
CA Algebra I: Slope and Y-intercept 27-32, figuring out the slope, y-intercept and equation of a line
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- We're on problem 27.
- And the question is, which equation best represents the
- graph above?
- So before even looking at the choices, let's see what we can
- figure out about the graph.
- So what's its y-intercept?
- So if we said that this is the equation of a line, we said y
- is equal to mx plus b where m is the slope and y is-- and b
- is the y-intercept.
- It's late, I'm getting slightly muddled.
- So what's the y-intercept?
- Well when x is equal to 0, y is equal to 0.
- So this is going to be equal to 0.
- The y-intercept is 0.
- When x is equal to 0, why is equal to 0.
- So the y intercept is 0.
- So we know that this has a form y is equal to mx where m
- is a slope.
- Let's figure out the slope.
- Slope is equal to change in y for a given change in x or
- change in y over change in x.
- So when we increase x by 1, how much are we increase-- or
- decrease y by?
- Well then y increases by 2.
- So we could say that equals y changes by 2, by plus 2 when x
- changes by plus 1.
- So we get the slope is equal to 2, so the equation of this
- line is y is equal to 2x.
- Which is choice B.
- Next problem.
- Which point lies on the line defined by 3x plus
- 6y is equal to 2?
- Well the best thing to do is just probably to substitute
- these numbers in for x and y and see which one works.
- So here x is 0, y is 2.
- So let's see.
- 3 times 0 plus 6 times 2 is equal to 0 plus 12.
- Doesn't equal 2, it equals 12.
- This one doesn't work.
- I'm just taking 3 times x plus 6 times y and
- seeing what it equals.
- In this case, we have 3 times 0 plus 6 times y.
- Plus 6 times 6.
- Well that's 0 plus 36.
- That doesn't equal 2.
- Can't be that choice.
- This one, we have 3, this 3, times 1.
- Plus 6 times this y.
- 6 times minus 1/6.
- So let's see.
- That's 3.
- That's equal to 3.
- And then, 6 times 1/6 is 1, but we have a
- negative out there.
- So it's minus 1.
- That is equal to 2.
- So that works.
- 3 times 1 plus 6 times negative 1/6 is equal to 2.
- So our answer is C.
- Problem 29.
- Let me see if I need to cut and paste this one in.
- All right.
- So yeah, I think it's a good idea.
- Let me see.
- Copy and paste it.
- Let me copy and paste the next several problems. We could
- just move-- be streamlined about it.
- OK, they all fit.
- Good enough.
- What is the equation of the line that has slope of 4 and
- passes through the point 3 comma minus 10?
- So if the slope is 4, we know that the line-- so just
- rewrite it in slope y-intercept form again.
- mx plus b.
- They're telling us that the slope is 4.
- So we know the equation of the line is y is equal to 4x plus
- the y-intercept.
- And then we could figure out the y-intercept by
- substituting in this point that they say it goes through.
- So it goes through the point 3 comma negative 10.
- So y is equal to minus 10 when x is equal to 3.
- So 4 times x.
- x is equal to 3.
- Plus b.
- So what is that?
- That's minus 10 is equal to 12 plus b.
- We could subtract 12 from both sides of this equation, and we
- get minus 22.
- Minus 10 minus 12 is minus 22.
- This 12 obviously goes away.
- Is equal to b.
- So the equation of our line is y is equal to 4x plus b, which
- we just figured out is minus 22.
- 4x minus 22.
- That's choice A.
- Problem 30.
- The date in the table show the cost of renting a bicycle by
- the hour, including a deposit.
- If hours, h, were graphed on the horizontal axis-- let me
- see if I can draw that.
- So that is on the horizontal axis I have hours, h.
- So this is h.
- And costs were graphed on the vertical axis, so let me draw
- the vertical axis.
- So cost is on the vertical axis.
- What would be the equation of the line that fits the data?
- OK, so they just want to know cost as a function of hours.
- So let's see.
- It's a linear relationship.
- We could treat this just like any other line.
- So if we write it as y equals mx plus b.
- I didn't even have to do this.
- So y is equal to mx plus b.
- So m, or the slope, is equal to change in y over-- or let
- me write it differently.
- Because we want it in terms of-- so we could write c is
- equal to the slope times the hours plus b.
- Instead of x, we have hours, and instead of y, we have the
- cost.
- So what's the slope?
- Slope is equal to change in the dependent variable, change
- in cost, divided by change in the independent variable,
- divided by change in hours.
- So let's see.
- If when our hours increased by 3, how much did our
- change in c go by?
- When we go from 2 to 5, so if we say 5 minus 2, that's how
- much we changed in hours.
- That's delta h.
- Then how much did we change in cost?
- Well it's 30 minus 15.
- So it equals 15 over 3.
- which is equal to 5.
- So we figured out the slope.
- Slope is equal to 5.
- Now we just have to figure out the y-intercept.
- So this equation, we could rewrite it as cost is equal to
- the slope, which is 5, times the hours plus whatever our
- y-intercept is.
- And then we just have to substitute one of the points
- in to solve for b, just like we did in the previous video.
- So let's substitute the point when h is equal to 2, c is
- equal to 15.
- So the point 2 comma 15.
- So when h is equal to 2, c is equal to 15.
- c is equal to 15 when h is equal to 2.
- And now we can just solve for b.
- So you get 15 is equal to 10 plus b.
- Subtract 10 from both sides, you get b is equal to what?
- 5.
- So the equation of this line is cost is equal to 5 times h
- plus our y-intercept, or plus our c-intercept we could say.
- So plus 5.
- 5h plus 5, and that is choice C.
- And if you try it out with these points,
- it should all work.
- OK, next problem.
- Let me see.
- I think this cutting and pasting works out well.
- I think I have time for at least two more.
- OK, let me clear this.
- Clear this one right here.
- Let me paste it in.
- OK, it says, some ordered pairs for a linear function of
- x are given in the table below.
- It's kind of like the last one.
- Which of the following equations was used to generate
- the table above?
- Well we could do the same thing we did the last time.
- We could say OK, this is going to have some relation.
- You know y is equal to mx plus b.
- It's going to be a line and this is the
- slope y-intercept form.
- And they tell us it's the linear function.
- So that's why we can immediately say that's going
- to be a line.
- There's no x squared or anything like that.
- So a good place to start is just with the slope.
- So slope is equal to change in y over change in x.
- So let's see.
- When x goes from 1 to 3, what does y go for.
- It goes from 1 to 7.
- So change in x is equal to 3 minus 1 or just 2.
- And what's change in y?
- 7 minus 1.
- So that is equal to 6 over-- oh, no.
- Sorry.
- Change in x was 3 minus 1.
- Sorry about that.
- 3 minus 1.
- So it's 6 over 2.
- When x changes by 2-- that's this 2, change in x-- y
- changes by 6.
- That's change in y.
- So the slope is 3.
- So the equation of our line is going to be y is equal to 3x
- plus some y-intercept.
- And now we could substitute some points in.
- Let's substitute this one.
- This 1, 1 seems easy.
- So y is equal to 1 when x is equal to 1.
- So we get 1 is equal to 3 plus b.
- Subtract 3 from both sides of this equation and we get what?
- 1 minus 3 is minus 2 is equal to-- obviously, this
- disappears, 3 minus 3.
- So b is equal to minus 2.
- So the equation of our line, we have a slope of 3 and a
- y-intercept of minus 2.
- The equation is going to be y is equal to 3x plus b.
- Well b is minus 2.
- So minus 2.
- And that is choice C.
- 32.
- The equation of line l is 6x plus 5y equals 3.
- And equation of line q is 5x minus 6y is equal to 0.
- Which of the statements about the two lines are true?
- OK, let's see.
- They have the same y intercept.
- They're parallel.
- They have the same x-intercept.
- Lines l and q are perpendicular.
- Well I'll tell you how you can identify perpendicular lines.
- They have the negative inverse of each other's slope.
- But I don't know if that's going to come into play yet.
- I always find it easier to look at these lines in kind of
- the slope y-intercept form.
- So let's do that.
- So line l can be 6x plus 5y is equal to 3.
- So if we were to subtract 6x from both sides you get 5y is
- equal to minus 6x plus 3.
- And then you divide both sides by 5 and you get y is equal to
- minus 6/5 x plus 3/5.
- So this is line l right here.
- I'll do line q in a different color.
- So q is 5x minus 6y is equal to 0.
- So let's subtract 5x from both sides.
- You get minus 6y is equal to minus 5x.
- If you divide both sides by negative 6, you get y is equal
- to-- minus 5 divided by negative 6.
- That's 5/6 x.
- Well, it actually looks like D comes into play.
- Because the slope of this one is minus 6/5.
- That's the slope of line l.
- The slope of line q is the inverse of that made negative.
- At this point it's a good thing to memorize.
- I haven't proven it to you yet.
- If you want a line that has a slope perpendicular to this
- one, or a line that is perpendicular to this one will
- have a negative inverse slope.
- So the inverse of this is minus 5/6 and the negative
- inverse is plus 5/6.
- So this is the negative inverse of that.
- So these lines are actually perpendicular.
- So that is D.
- See you in the next video.
Be specific, and indicate a time in the video:
At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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