### All SAT Math Resources

## Example Questions

### Example Question #63 : Systems Of Equations

Evaluate .

**Possible Answers:**

The system has no solution.

**Correct answer:**

Set up the system:

Multiply both sides of the first equation by 5, and both sides of the second by :

Add both sides of the equations, then solve for in the resulting equation:

### Example Question #64 : Systems Of Equations

Evaluate .

**Possible Answers:**

The system has no solution.

**Correct answer:**

In the equation , substitute for :

Simplify, then solve for by isolating the variable:

### Example Question #281 : Algebra

Evaluate .

**Possible Answers:**

The system has no solution.

**Correct answer:**

Multiply both sides of each equation by the least common denominators of their coefficients in order to make those coefficients into whole numbers:

, so

, so

This sets up the system

Multiply the latter equation by on both sides, then add to the former equation to eliminate the terms:

Solve for :

### Example Question #281 : Algebra

Evaluate .

**Possible Answers:**

The system has no solution.

**Correct answer:**

In the equation , substitute for :

Simplify the left expression:

Now solve for :

Substitute this value in the following equation and evaluate for :

### Example Question #281 : Equations / Inequalities

A system of two linear equations has no solution.

One of the equations is

.

Which of the following *could* be the other equation?

**Possible Answers:**

None of the other choices gives a correct answer.

**Correct answer:**

One way to solve this is to know that the two equations of a linear system without a solution - an inconsistent system - are represented by a pair of parallel lines, which are lines with the same slope. To find the slope of the line of the equation , rewrite the equation in the slope-intercept form , with the slope of the line. This is done by solving for , so isolate in the equation as follows:

The slope of the line is , the coefficient of .

The equations in all four choices are already in slope-intercept form, so examine the coefficient of in each. The equation is the only one with -coefficient , so its slope is also . Since the lines of the equations are of the same slope, they are parallel, and the equations form a system of equations with no solution. The correct response is therefore .

### Example Question #61 : Systems Of Equations

If and , find the value of .

**Possible Answers:**

**Correct answer:**

There are three variables, but only 2 equations. With fewer equations than variables, it is not possible to solve for the value of each variable separately, but we can eliminate variables to manipulate the equation into the form we want.

Multiply the second equation by on both sides, resulting in .

Then add the equations together (allowed since we are adding the same thing to both sides, just in a different form):

Simplify:

Resist the urge to keep working, since that's exactly what they've asked for. Done!

### Example Question #71 : How To Find The Solution For A System Of Equations

At what point on a graph do these lines intersect.

**Possible Answers:**

**Correct answer:**

Use the addition method to eliminate one variable and therefore solve the system of equations.

__________________

Now substitute to find x.

### Example Question #284 : Algebra

A farmer sells only apples and oranges at his stand. Apples are $0.55 each and oranges are $0.65 each. If he makes $33.45 in one day after selling 55 total pieces of fruit, how many of each fruit did he sell?

**Possible Answers:**

32 oranges and 21 apples

32 oranges and 23 apples

25 oranges and 30 apples

47 oranges and 23 apples

**Correct answer:**

32 oranges and 23 apples

Translate the word problem into a system of equations:

Use the first equation to solve for A in terms of O, or vice versa.

Use substitution to solve for O in the second equation.

Now plug the value for O back into the first equation to solve for A.

### Example Question #281 : Equations / Inequalities

If , what does equal?

**Possible Answers:**

**Correct answer:**

Subtract and from the both sides to get .

Divide both sides by , to get

### Example Question #282 : Equations / Inequalities

The graph below is the graph of a piece-wise function in some interval. Identify, in interval notation, the decreasing interval.

**Possible Answers:**

**Correct answer:**

As is clear from the graph, in the interval between ( included) to , the is constant at and then from ( not included) to ( not included), the is a decreasing function.