# GMAT Math : Algebra

## Example Questions

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### Example Question #1 : Solving Linear Equations With Two Unknowns

Which equation is linear?

none of them are linear

Explanation:

Let's go through all of the answer choices.

1. and are both constants, so the equation is actually linear.

2. 5x + 7y - 8yz = 16: This is not linear because of the yz term.

3. : This can be transformed into y + 8 = (x + 6)(x - 2).  Clearly when this is expanded, there will be an term, so this is not linear.

4. : This is not linear either, also because of the term.

### Example Question #2 : Solving Linear Equations With Two Unknowns

Solve.

Explanation:

Solve for in the first equation:

Substitute into the second equation:

Solve for .

### Example Question #2 : Solving Linear Equations With Two Unknowns

What is

Explanation:

Solve the first equation to get

Substitute that into the second equation and get

Solve the equation to get , then substitute that into the first equation to get .

Plugging those two values into , gives

### Example Question #4 : Solving Linear Equations With Two Unknowns

Solve the system of equations:

The system has no solution.

The system has no solution.

Explanation:

Multiply both sides of the first equation by 12:

Now, add both sides of the two equations:

Since this is impossible, the system of equations is inconsistent and thus has no solution.

### Example Question #5 : Solving Linear Equations With Two Unknowns

Give the solution set for .

Insufficient information is given to answer the question.

Explanation:

The expression on the left factors as the difference of squares:

Since , we can substitute:

We now have a system of linear equations to solve:

### Example Question #6 : Solving Linear Equations With Two Unknowns

A company wants to ship some widgets.  If the weight of the box plus one widget is 6 pounds, and the weight of the box plus two widgets is 10 pounds, then what is the weight of the box and the weight of the widget?  Put the answer in an ordered pair such that the ordered pair is (box weight, widget weight).

Explanation:

Let the weight of the box be represented by  and the weight of the widget be represented by .  Since the weight of the box plus the weight of one widget is 6 pounds, this can be represented by the equation

Since the weight of the box plus two widgets is 10 pounds, this can be represented by the equation

We now have two equations and two unknowns and we can now solve for  and .  To do this we solve the first equation for  and substitute it into the second equation.  Solving the first equation for  we get

Substituting this into the second equation we get

Using  and substituting it into the first equation we get

So the weight of the box is 2 pounds and the weight of the widget is 4 pounds.  This gives us the ordered pair .

### Example Question #7 : Solving Linear Equations With Two Unknowns

Solve for  when

Explanation:

Plug in the given value and then isolate .

### Example Question #8 : Solving Linear Equations With Two Unknowns

Whats the value of  when :

Explanation:

Solve for :

Explanation:

What is ?