### All SAT Mathematics Resources

## Example Questions

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

If and , what is the value of ?

**Possible Answers:**

7

9

5

11

**Correct answer:**

7

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 ?

**Possible Answers:**

2

3

4

5

**Correct answer:**

2

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 ?

**Possible Answers:**

4

25

20

5

**Correct answer:**

20

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 ?

**Possible Answers:**

13

11

15

9

**Correct answer:**

13

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?

**Possible Answers:**

-3

-2

2

3

**Correct answer:**

-3

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 ?

**Possible Answers:**

-3

3

-2

0

**Correct answer:**

-2

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 ?

**Possible Answers:**

13

4

10

7

**Correct answer:**

4

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 ?

**Possible Answers:**

2

4

9

7

**Correct answer:**

2

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?

**Possible Answers:**

**Correct answer:**

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?

**Possible Answers:**

**Correct answer:**

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 .

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