CAHSEE practice: Problems 47-51

Problem 47. Which graph represents the system of equations shown below? So each of these equations represents a line. So this is going to be a line, well, both of them have y-intercepts of plus 3. And this line has a slope of minus 1. So let's see. We need a downward sloping thing that intersects at positive 3. Actually, they're both going to intersect at positive 3. So choice A, neither of them. Both of their y-intercepts are minus 3. We need to see a plus 3 here. So it's not choice A. In choice C, one of the lines intersects at positive 3, but the other line intersects at minus 3. And they also have the same slope, and these guys obviously don't have the same slope. So it's not going to be choice C. Choice B. So I have two lines. Both of them have a y-intercept of 3. This looks good so far. Both of them have y-intercepts of 3. And then I have one line that has a positive slope of 1. For every 1 I go over, I go up 1. And I have one that has a negative slope of 1. When I move over 1, I go down 1. This line right here is y is equal to x, because I have a slope of 1. 1x plus 3. 3 is its y-intercept. And this line right here is y is equal to-- its slope is minus 1. You move to the right 1, you go down 1. So minus 1x, or minus x, and its y-intercept is plus 3. So our answer is B. These are the two equations that we have up there. You can't see it yet. That we have right here. Our answer is B. And D, both of these guys have the same slope, so you can't have-- clearly it doesn't apply to the two equations up there. And they also have two different y-intercepts, so it's definitely not choice D. Problem 48. Yoshi has exactly $1 in dimes, which is $0.10, and nickels, $0.05. If Yoshi has twice as many dimes as nickels, how many nickels does she have? So she has $1. So let n equal the number of nickels, and let d is equal to the number of the dimes. So it says Yoshi has twice as many dimes as nickels. So the dimes are twice as many as the number of nickels. That's what this tells us. Yoshi has twice as many dimes as nickels. That translates into the dimes are 2 times the nickels. And then we also know that she has $1. So how do you get $1? Well, it's $0.10, so 0.1 times the number of dimes. Plus 0.05 times the number of nickels is equal to $1. Or maybe just so we can avoid decimals, I could rewrite this statement right here, exactly $1. Instead of writing in dollars, I could write in cents. I could write 10 cents times each dime, plus 5 cents times each nickel is equal to 100 cents. So if you compare this equation and this equation, I just multiplied both sides by 100. Here I'm dealing with cents-- $1 is 100 cents. Here I'm dealing with dollars. A dime is 1/10 of a dollar. A dime is 10 cents. This one right here is probably a little bit easier to deal with. So what are they asking? How many nickels does she have? So we want to solve for n, we want to solve for nickels. We have this piece of information right here. The dimes are equal to 2 times the number of nickels. And we have this right here. So what we can do is substitute for d. We can replace this d, and we know that d is equal to 2n. So we could put 2n in the place of that. So let me write it over here. So I have 10d plus 5n is equal to 100. 10 cents times my number of dimes plus 5 cents times my number of nickels is equal to 100 cents. And then my dimes is equal to 2n. So 10 times 2n plus 5n is equal to 100. 10 times 2n is just 20n, plus 5n, is equal to 100. And then we get 25n is equal to 100. Divide both sides of this equation by 25, and you get n is equal to 100 divided by 25 is 4. Yoshi has 4 nickels. And if you want to know how many dimes, you multiply it by 2. She has 8 dimes. So she has $0.80 in dimes and she has $0.20 in nickels, 4 times 5. Problem 49. What are all the possible values of x such that 10 times the absolute value of x is equal to 2.5? So we can simplify this. So we have 10 times the absolute value of x is equal to 2.5. Or if we divide both sides of this equation by 10, we get-- let me do it in blue. We get the absolute value of x is equal to-- what's 2.5 divided by 10? You might be able to just do it in your head. You say, oh, you know, there's 4 quarters in $1, or 25 times 4 is 100. But this is 25 divided by 10. You multiply 2.5 times 4, you're going to get 10. So you can say this is 1/4. If that wasn't obvious to you, you could actually do the division. You can divide 10 into 2.5. Maybe throw some 0's there. Put the decimal there. 10 goes into 2 zero times. 0 times 10 is 0. Bring down the 2. 2 minus 0 is 2. Bring down the 5. 10 goes into 25 how many times? 10 goes into 25 two times. 2 times 10 is 20. 25 minus 20 is 5. Bring down another 0. 10 goes into 50 exactly five times. 5 times 10 is 50. Subtract, you get 0. So 10 goes into 2.5, 0.25 times, which is the same thing as 1/4. The absolute value of x is equal to 0.25. Now that tells us that x, what are the two values for x? x could definitely be equal to 0.25, because the absolute value of 0.25 is 0.25. Or what other number, when I take its absolute value, is 0.25? Well, it could be minus 0.25. I take the absolute value of minus 0.25, I also get 0.25. So my two choices are 0.25 or minus 0.25, and that is choice A right there. And when we did this division, I did it kind of a long and slow way. And if you didn't recognize this as 1/4. The other way to say it is when you divide by 10, that's equivalent to moving your decimal 1 slot to the left. So you could have just said, oh, 2.5 divided by 10 is equal to-- let's just move the decimal over to the left-- is equal to 0.25. All of those would have been equivalent ways of doing it. Problem 50. Which of the following is equivalent to 1 minus 2x is greater than 3 times x minus 2? So let's solve this. So we have 1 minus 2x is greater than 3 times x minus 2. Or we get 1 minus 2x is greater than-- let's distribute our 3-- 3x minus 6. 3 times minus 2 is minus 6. Now we can subtract 1 from both sides. So we have a minus 1 plus here, and we're going to subtract a 1 from that side. The whole reason why I'm doing that is to cancel out that 1 right there. So this side, those cancel out, so I'm just left with minus 2x is greater than 3x, and then minus 6 minus 1 is minus 7. And now here's the vaguely tricky part. No, actually, we're not at the vaguely tricky part yet. Let's subtract 3x from both sides. So we have minus 3x on that side, and then minus 3x on that side. And the whole reason why I'm doing that is to cancel out this 3x. So that these two guys obviously cancel out. Now minus 3x plus minus 2x, which is minus 5x, is greater than minus 7. And now here's the vaguely tricky part. We want to divide both sides by minus 5. But when you have an inequality, and you're dividing or multiplying both sides by a negative number, you need to switch the inequality. So if we divide both sides by minus 5, so I could multiply by a fraction, minus 1/5 times minus 5x. And then I'm going to have a minus 7 times minus 1/5. I'm going to switch the inequality. So it goes from greater than to less than. And the whole reason why it's switched is because I'm multiplying both sides, or depending on how you do it, dividing both sides by a negative number. So then this becomes-- I picked minus 1/5 so this cancels out. So this just becomes x is less than, and then minus 7 times minus 1/5, the negatives cancel out, or a negative times a negative is a positive. So you're left with 7/5. So x is less than 7/5. And it shows you that maybe you don't want to go work out the whole problem. Because right here, they didn't expect me to figure out the whole problem, so I just wasted a little bit of both of our time. They just wanted us to go a couple of steps into it. So what did they just do? Well, they just wanted that first step, where I distributed the 3. 3x minus 6. So that's all they did here. Because they didn't change the left-hand side anywhere, they didn't change the inequality anywhere. So all I have to do is pick choice C right there. But hopefully you learned a little bit about inequalities as well. I hate to waste your time like that, but this was actually a fairly straightforward problem. But actually, the very first step we did got us to the answer. So that teaches us both a lesson: look at the choices before you proceed with the problem. 51. Which equation represents the line on the graph below? So we can look at this, we can just kind of eyeball it. But sometime it's hard to eyeball it. So what we want to do is look for some point, my first intuition is to say, OK, y is equal to mx plus b. Let's look for the y-intercept and the slope. But here the y-intercept, it's not completely obvious where it is. Although it looks like it's at 1.5. And we can actually verify that it's at 1.5, because its slope, you can see right here. If we start at 1, and if we go 2 to the right, we go down by exactly 1. So its slope, m, is equal to change in y over change in x. When your change in x is positive 2, your change in y is minus 1. So your slope is minus 1/2. So if you go back 1, you're going to go up 1/2. Just like that. So our y-intercept is 1.5. It's right there. It's equal to 1.5. Just like that. I can verify that from the slope. If I start at 2, and I go 1 to the right, that means I'm going to go 1/2 down. Because my slope is minus 1/2. So we go 1/2 down, and that'll get me to 1.5. So the equation of this line is y is equal to minus 1/2 x, that's my slope, plus 1.5. And that doesn't look like any of these choices here, so we need to simplify this a little bit. Well, the first thing we might want to do is multiply both sides of this equation by 2. So you get 2y is equal to 2 times minus 1/2 x, is just minus x, and then 2 times 1.5 is 3. Plus 3. And then if we add x to both sides of this equation, and I'm doing that to get rid of this minus x on the right-hand side, we get x plus 2y is equal to 3. And that is choice A right there, which was completely equivalent to what we did up here.