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: 

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

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:    

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:  

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.

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 .

Equation 1:

Equation 2:

Equation 3:

Adding the terms of the first and second equations together will yield .

Then, add that to the third equation so that the y and z terms are eliminated. You will get .

This tells us that x = 1. Plug this x = 1 back into the systems of equations.

Now, we can do the rest of the problem by using the substitution method. We'll take the third equation and use it to solve for y.

Plug this y-equation into the first equation (or second equation; it doesn't matter) to solve for z.

We can use this z value to find y

So the solution set is x = 1, y = 2, and z = –5/3.

Step 1: Multiply first equation by −2 and add the result to the second equation. The result is:

  

Step 2: Multiply first equation by −3 and add the result to the third equation. The result is:

Step 3: Multiply second equation by −23 and add the result to the third equation. The result is:

Step 6: solve for x by substituting y=2 and z=1 into the first equation.

To solve the equations we want to try to cancel out some of the letters to reduce the equations. We can do this by adding different equations to each other. If we add:

we get a new equation, 

.

We can also add 

to get .

So we can rearrange the new equations to get

and substitute these into the original second line equation:

Then we can solve for b and c:

Add the first and third equations together, to eliminate z:

+ 

This yields the following: 

Divide out a 2, to simplify the equation: 

Plug this value back into the first equation: 

Solve for z:

Add the first and second equations together to eliminate y:

+

This yields the following: 

Divide out a 3 to simplify the equation: 

Plug in the value for z:

Solve for x: 

Plug the values for x and z into the first equation: 

Simplify: 

Solve for y:

Make sure the values for x,y, and z satisfy all of the equations:

Solution: