If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

## Digital SAT Math

### 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 •  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.
• 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. • At , why does Sal take the reciprocals and swap both sides instead of just cross multiplying? • I'm having trouble understanding this whole topic. Can someone tell me what slope exactly means? • 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. • 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.
• 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. • how could you write so well with the mouse • , shouldnt the denominator be 18 • You could have gotten the same answer by using the formula a1/a2=b1/b2 faster than using the above slope formula.  