College Algebra : Linear Systems with Two Variables

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Example Question #11 : Linear Systems With Two Variables

Solve this linear system:  

Possible Answers:

No solution

Correct answer:

Explanation:

To solve this linear system we must find the coordinate point where both lines intersect. To do so, we eliminate one variable (by multiplying or dividing by some amount) and solve for the remaining variable.

1) 

2) 

Multiply equation 1 by 5 and equation 2 by 6; add the two equations to cancel out the x term:

1) 

2) 

Solve for y:

Now substitute this value back into either original equation to get the y-coordinate:

Rounded answer: 

 

 

Example Question #11 : Systems Of Equations

Solve the system of equation using elimination:

Possible Answers:

Correct answer:

Explanation:

To solve by elimination, we want to cancel out either the  or  variable:

 

 

 

Now that we know the value of , we can plug it in to one of the equation sets to solve for :

We can then conclude that

Example Question #13 : Linear Systems With Two Variables

Solve the following system of equations:

        

Possible Answers:

  

   

   

  

Correct answer:

  

Explanation:

   

 

Solve by adding the two equations: 

                                                    

Adding the two equations yields the following: 

Dividing both sides of the equation by 3 yields the following: 

Plug value of x back into first equation: 

Subtracting 3 from both sides of the equation yields the following: 

Make sure the values for x and y satisfy both equations:

          

Solution:    

Example Question #14 : Linear Systems With Two Variables

Solve for :  

Possible Answers:

Correct answer:

Explanation:

We can evaluate the value of  by subtracting the first equation from the second equation since both equations share , and can be eliminated.

The equation becomes:  

Substitute this value back into either the first or second equation, and solve for y.

The answer is:  

Example Question #15 : Linear Systems With Two Variables

Consider the system of linear equations

True or false: This system has one and only one solution.

Possible Answers:

True

False

Correct answer:

True

Explanation:

The given system has more equations than variables, which makes it possible to have exactly one solution.

One way to identify the solution set is to use Gauss-Jordan elimination on the augmented coefficient matrix

Perform operations on the rows, with the object of rendering this matrix in reduced row-echelon form.

First, a 1 is wanted in Row 1, Column 1. This is already the case, so 0's are wanted elsewhere in Column 1. Do this using the row operations:

An all zero row has been created, so it must be moved to the bottom:

Now, get a 1 into Row 2, Column 2:

Now, get 0's in the other positions in Column 2:

This matrix is in reduced row-echelon form, and can be interpreted to mean that

.

Thus, the system of equations given has one and only one solution.

 

Example Question #11 : Systems Of Equations

Solve this system of equations:

Possible Answers:

None of these

Correct answer:

Explanation:

To solve this system of equations we must first eliminate one variable and solve for the remaining variable. We then substitute the variable back into our original equation and solve for the second variable still unknown.

We will use the elimination method:

Multiply the top equation by 1 and add it to the second equation:

--------------------------

Now we substitute the value of x into our original equation:

Thus, our lines are equal (intersect) at .

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