# SAT Mathematics : Linear Algebra

## Example Questions

### Example Question #1 : Solving Systems Of Equations

If  and , what is the value of ?

7

9

5

11

7

Explanation:

When you're facing systems of equations, the SAT often provides you a shortcut if you recognize it. When a systems of equation question asks you to solve for a combination of variables (e.g. ) and not just a single variable, there's often a faster way to solve directly for the combination.

Here, recognize that if you simply add the two equations together - much like using the "Elimination Method" but without actually trying to eliminate a variable - you can get  and  together with the same coefficient.

sums to:

If you then divide both sides by 8, you can get to exactly the answer they're looking for:

### Example Question #2 : Solving Systems Of Equations

For the system of equations above, what is the value of ?

2

3

4

5

2

Explanation:

This system of equations provides you an excellent opportunity to use the Elimination Method to isolate a single variable. With the two equations provided, you already have a negative  term (in the second equation) and a positive  term (in the first).  If you multiply the second equation by 2, you can get the coefficients the same and sum the equations to arrive at a single variable, . First, multiply the entire second equation by 2:

Then stack the updated equations and sum them:

Gives you:

This means that  but remember to always double check that you've solved for the proper variable. The question wants to know  so you can plug  back into either equation to solve. Using the original second equation:

So

### Example Question #2 : Solving Systems Of Equations

If   and , what is the value of ?

4

25

20

5

20

Explanation:

This problem asks you to solve for , and conveniently provides you with a single-variable equation that allows you to solve directly for  so that you're halfway home already:

, so if you divide both sides by  you'll find that .

You can then plug in  to the first equation to get:

Adding  to both sides gives you:

And then divide both sides by  to get .

You now have your values for  and  so you can multiply them to get :

, making  the correct answer.

### Example Question #3 : Solving Systems Of Equations

For the system of equations above, what is the value of ?

13

11

15

9

13

Explanation:

This problem affords you a quick opportunity to use the Elimination Method. The first equation has a positive  term and the second has a negative  term, meaning that if you sum the two equations you will eliminate the  terms and be left with a single variable, .

The two equations sum to:

And dividing both sides by  allows you to determine that .

Note that the question asks for , not , so you need to plug  back in to one of the two equations to solve for . Using the first equation, you have:

So  meaning that .

### Example Question #5 : Solving Systems Of Equations

In the system of equations above,  is a constant. For which of the following values of  does the system have no solution?

-3

-2

2

3

-3

Explanation:

One way to look at a system of linear equations is that the solution to that system is the point at which the graphs of the lines intersect at the same  point. So a system of linear equations WILL NOT have any solutions if the lines never meet; in other words, if the lines are parallel with different y-intercepts.

The "by the book" method to determine if lines are parallel is to put each into slope-intercept form, , and then see if the slopes  are the same. For the first line, that's:

So for the second line, you would need to find the  value that makes the slope equal to -2. To get closer to slope-intercept form of the second equation you can start at:

And then divide both sides by  to isolate the  term:

If  you've matched the slope of , making  the correct answer.

Of course, there's a shortcut to this. If you recognize that between the two equations, the scale factor from first to second is that  is multiplied by 3, you can choose a value for  that provides the same effect for the  term. Since  is multiplied by  in the second equation, you can say that:

So .

### Example Question #4 : Solving Systems Of Equations

For the system of equations above, what is the value of ?

-3

3

-2

0

-2

Explanation:

For this problem, there are reasons to choose either the Elimination Method or the Substitution Method to solve the system. The question asks for the value of  and the first equation gives you a great opportunity to substitute for  in terms of  and use the Substitution Method.  Since , you can conclude that:

And then substitute  where  appears in the second equation:

Distribute the multiplication across parentheses to get:

And then solve:

Of course, you could also use the Elimination Method. If you multiply the first equation by -3, you'd get:

, which you can stack with the second equation and sum:

When you sum, the  terms cancel leaving you with the answer:

### Example Question #5 : Solving Systems Of Equations

In the system of equations above, what is the value of ?

13

4

10

7

4

Explanation:

This system of equation gives you an opportunity to use the Elimination Method to quickly eliminate the variable  and then use a single-variable equation to solve for . If you multiply the first equation by 3, you will then have a  term in the first equation and a  term in the second, so adding the two equations will eliminate the .

Once you've multiplied the first equation by 3, you'll sum the equations:

Adding these together gives you:

And then you can solve for  by multiplying each side of the equation by  to get .

Note that you should always double check that you've solved for the right variable (or combination of variables) for the question. This question does ask for  so your answer is .

### Example Question #3 : Solving Systems Of Equations

If  and , what is the value of ?

2

4

9

7

2

Explanation:

You can subtract the second equation from the first equation to eliminate :

You could also solve one equation for  and substitute that value in for  in the other equation:

### Example Question #6 : Solving Systems Of Equations

Which ordered pair  satisfies the system of equations above?

Explanation:

Whenever you have a chance to solve a system of equations using the Elimination Method, it is generally your fastest option. Here if you multiply the first equation by 2, you can then subtract the two equations:

gives you a new first equation of .  When you then stack and subtract:

You're left with:

So .

Then just plug  back in to one of the two equations and you can solve for . If you use the first, you'll have:

So .  This means that the ordered pair is .

### Example Question #7 : Solving Systems Of Equations

Which of the following ordered pairs satisfies the system of equations above?

Explanation:

This problem provides you with a quick opportunity to use the Elimination Method. Because there is an  in the first equation and  in the second, if you add the two equations you can eliminate the  terms and solve straight for :

This means that .

Then you can plug  into either of the equations and you'll solve for . Using the first equation, that would be:

So

This means that the correct ordered pair is .