How to solve two equations with two unknowns

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Which equation is linear?

Correct answer:

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.

Solve.

Correct answer:

Explanation:

Solve for in the first equation:

Substitute into the second equation:

Solve for .

Correct answer:

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

Solve the system of equations:

Possible Answers:

The system has no solution.

Correct answer:

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.

Give the solution set for .

Possible Answers:

Insufficient information is given to answer the question.

Correct answer:

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:

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).   

Correct answer:

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 .

Correct answer:

Explanation:

Plug in the given value and then isolate .

Whats the value of  when :

Correct answer:

Solve for :

Correct answer:

Correct answer:

Explanation:

From the second equation:

Substitute into the first, then solve:

All GMAT Math Resources

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