Main content

### Course: Digital SAT Math > Unit 2

Lesson 5: Solving systems of linear equations: foundations# Solving systems of linear equations — Harder example

Watch Sal work through a harder Solving systems of linear equations problem.

## Want to join the conversation?

- I wish they would tell us if this particular question can be solved with or without a calculator bc on the real sat there are 23 math section ... 1 with and 1 without calc so if Sal would tell us if the question he's solving is calc or no calc it would be easier to figure a strategy for the real SAT(53 votes)
- A calculator would not really help you much with this math. The calculator cannot do the rearrangement of the equations nor pick out the key items that have to be equal in the two different equations. The question is checking your understanding of what is necessary in a system of equations for there to be no solutions, and then asking you to calculate a factor that would cause the system to fit that category.

In fact, this**could**be on the part of the SAT where a calculator is allowed, but a calculator would get in the way. You have so many variables at once and fractions--the calculator would be throwing decimals back at you on the few calculations without variables. I recommend saving a calculator for problems that really need it, and those are rare, even on the SAT. So many can be conquered with factoring and canceling and simplifying, or recognizing special cases.(36 votes)

- suggestion: The "Harder example" could be harder than this. I am saying this because when we go to the "practice" exercises the problems are more difficult and have more types of question of the same subject, so it would be better if the examples was more complexy or maybe has more than only two examples. Because obviosly the subject is not limited of these only two types of question that the videos illustrate.(80 votes)
- Im lost right now(47 votes)
- At3:04, why does Sal take the reciprocals and swap both sides instead of just cross multiplying?(20 votes)
- ok this is the first and only video I didnt understand. what??(9 votes)
- So just to start,
**don't**waste your time converting the equations to slope intercept form! That's awaste of time!**huge**

Ok, now let's start :)

If you have two parallel lines (no solutions) than in the ax + by = c form, a and b are going to be the same, but c has to be different. Because if a, b,*and*c were the same, you'd have the same line, which would give you infinitely many solutions.

Ok, so we know that the coefficients of x and y have to be the same. Well, let's start with what we know. Let's make the first equation the same as the second equation. To get from 9 to 2 in the first equation (for the x coefficients) we divided by 4.5, right?

Ok, well let's use that same standard to get from 14 to a (for the y coefficients). Let's divide 14 by 4.5. And what is the answer? 3.111111111111111

So we know that a is 3.1111111111111111 (the 1's go on forever). Now which of these choices represents that? We don't even have to work them out on our calculator. The first two are negative so we don't want those. 9/14 is obviously less than 1 and we want something greater than 1 (3.111111111), so we're left with 28/9, which is obviously greater than 1 and if you actually work it out on a calculator you get 3.1111111111111.

Ok, I super hoped this helped and made it a little easier to understand! And I hope I wasn't too late!(34 votes)

- I'm having trouble understanding this whole topic. Can someone tell me what slope exactly means?(0 votes)
- slope is steepiness of line , in other words we can say that slope is rise/run.

vertical line have infinite slope while horizental line have zero slope(12 votes)

- An easier way to do this would be to make both equations have the same slope in its current notation (no solution is same slope and different intercept). You do this by finding a common multiple between the two to get 18x - 28y = -6 and 18x -9ay = -54. Because the slopes are equal and this is no solution, -9ay is -28y. divide by 9 to find a, which will give you 28/9.(13 votes)
- The only advice I can give you is to practice. it is the best way to learn new things, and you can always solve what you are stuck on eventually.(0 votes)

- Slope in math is defined as the rate of change of the y variable as x changes. This simply means that if you have one variable that is dependent (y) on an independent variable (x), then the slope is given as change in y divided by change in x. Let's say that we have a graph of distance (y) and time (x). We know that distance is dependent because time determines how much distance something will travel. Assuming the graph is a straight line, if we want to find the slope we would find the difference between 2 y values on an interval of x values. We would divide that difference by the difference between the interval of x values we used. For example, if a car is at 10 miles from a house in 10 min and at 20 miles from a house at 15 min, then our interval of x values is 10 min - 15 min. Slope is change in y divided by change in x. On that interval, the change in y is (20 miles - 10 miles) and change in x is (15 min - 10 min). Slope is then calculated as 10miles / 5 minutes or 2 miles/minutes. From the slope, we can see that every minute or change in x by 1, the car is traveling 2 miles or changing y by 2 miles.

Hope this was helpful.(8 votes)- The equation gives GALLONS left in her car.

It says that the number of gallons left equals the total number held by the gas tank (15 gallons) minus a strange expression, m/32

So, logic tells us that

Gallons left = Starting gallons minus gallons used

or

Gallons left = 15 - gallons used

In the gallons used spot is m/32

Cool, so how do we figure out gallons used? All we know is that she drives m miles and we are told in the formula we are given that in the spot where we need gallons used we have miles divided by 32 somethings

That is where Sal has miles/units = gallons

We can use the rules of algebra to isolate for the units to figure out WHAT they are:

miles/units = gallons

move the units out of the denominator by multiplying both sides by units

units x miles/units = gallons x units

cancel out units on the left:

miles = gallons x units

isolate the units by dividing both sides by gallons

miles/gallons = gallons x units /gallons

gallons cancel out on the right

We discover that the units of the number in the denominator are miles per gallon (mpg)

The only thing the question asks for is what the meaning of that 32 is in the denominator, and we just discovered that the units are miles per gallon. So, its meaning is 32 miles to the gallon, which is a different way of saying 32 miles per gallon.

Now, does that make sense?

Well, every mile driven will use a little bit of gas (hopefully a VERY little bit of gas). If the car gets 32 miles of driving for every gallon, then one 32nd of every gallon is used every time the car drives a mile. That is 1/32 gallons used for every mile driven.

Whew, I think that was the reciprocal you wanted to know about.(9 votes)

- I hardly understand this section...

I need more(12 votes)- I would check out other courses on Khan Academy (such as Algebra 1), and see if you can find more information there. Also, to get by the Digital SAT, you could just put the equation into the Desmos graphing calculator and test all the choices.

I hope this helps! :D(1 vote)

- If your unsure on your answer, you could plug it into Desmos and see what the line looks like (only IF your unsure, using the calculator might seem like the go to method but sometimes it's faster to solve then plugging it in / getting it solved)(6 votes)

## Video transcript

- [Instructor] Consider the system of linear equations above. Which of the following
choices of a will result in a system of equations
with no solutions? No solutions. So a system has no
solutions is if both lines and these are both linear equations, they actually tell us
these are linear equations, is if you have two
lines that are parallel, then you have no solutions. They are never going to intersect. There's not xy pair that
satisfies both of them. So if they are parallel, if they are parallel, then you're going to have no solutions. So what makes two lines parallel? Well if they have the same slope but they have different y-intercepts, they have different y-intercepts, you're going to be parallel. So essentially we need to pick the a here that makes the second line
parallel to the first. And to do that I'm
actually write both of them in kind of the slope intercept form where y is equal to the slope
times x plus the intercept. So let's do that. Let's do that for both of these. So first I'll do this one up here. So you have 9x minus 14y is equal to -3. See if I subtract 9x from both sides I would get -14y is
equal to -9x minus three. See if I divide both sides by negative 14 I'm going to get y is equal to negative 9 divided by negative 14 is positive nine over 14x and then plus three over 14. So that's this line written
in slope intercept form. Now let me write the second
line in slope intercept form. So I have 2x minus ay is equal to -6 and let's see if I subtract
two from both sides, I get -ay is equal to -2x. I'm subtracting 2x from both sides so I get minus 2x minus six and
then I can divide both sides by negative a and I get
y is equal to -2 divided by negative a is positive two over a x, and then plus six over six over a. Alright so we need to set up a situation, we need to set up a
situation where two over a is equal to nine over 14. These two things have
to have the same slope. And then when we're able
to figure out that a we have to verify that they
have different y-intercepts 'cause if they have the same
slope and the same y-intercept instead of having no solutions, they would have an infinite
number of solutions 'cause then it would be the same line. But let's solve for a. So we know that nine over 14 we know that nine over 14 needs
to be equal to two over a. Two over a, or another way
of thinking about this, there's a bunch of sometimes
people say cross multiply and all of that. I like to just do logical
algebraic operations. But we can take the
reciprocal of both sides so we could have, and actually
let me swap both sides. So we could say a over two
is equal to 14 over nine. Is equal to 14 over nine. And then multiply both sides by two. Multiply both sides by
two, and you're gonna get a is equal to 28 over nine. A is going to be this right over here. This is 28 over nine. So if I'm under time pressure
I already see a choice that's looking pretty good. But if we really want to care that we've, if a is 28 over nine, these two things are going to have the same slope. But let's make sure they don't
have the same y-intercept and I wouldn't do this if
I was under time pressure. But just for to feel good about it, if this y-intercept is six over a. So it's going to be six over 28 over nine, which is six divided by a
fraction is the same thing dividing by a fraction is the same thing as multiplying by its reciprocal, so it's gonna be the same thing
as six times nine over 28. And this is going to be
equal to six times nine is 54 over 28, which is clearly
different than three over 14 so if a is 28 over nine, same
slope, different y-intercepts and you're dealing with parallel lines. They will not intersect and you're going to have no solutions.