Solve Systems of Linear Equations - Pre-Calculus

In order to solve a system of linear equations, we must start by solving one of the equations for a single variable:

We can now substitute this value for y into the other equation and solve for x:

Our last step is to plug this value of x into either equation to find y:

We first move to the left side of the equation:

Subtract the bottom equation from the top one:

Left Side:

Right Side:

So

So dividing by a -1 we get our result.

For any system of linear equations, we can start by solving one equation for one of the variables, and then plug its value into the other equation. In this system, however, we can see that both equations are equal to y, so we can set them equal to each other:

Now we can plug this value for x back into either equation to solve for y:

So the solutions to the system, where the lines intersect, is at the following point:

In order to solve a system of linear equations, we can either solve one equation for one of the variables, and then substitute its value into the other equation, or we can solve both equations for the same variable so that we can set them equal to each other. Let's solve both equations for y so that we can set them equal to each other:

Now we just plug our value for x back into either equation to find y:

So the solution to the system is the point:

Start from equation 3 because it has the least number of variables. We see directly that .

Back substitute into the equation with the next fewest variables, equation 2. Then,

. Solving for , we get

or .

Then back substitute our and into equation 1 to get

.

Solving for x,

.

So our solution to the system is

We can solve the system using elimination. We can eliminate our by multiplying the top equation by :

and then adding it to the bottom equation:

                                          

We can now plug in our y-value into the top equation and solve for our x-value:

Our solution is then

We can solve the system using substitution since the bottom equation is already solved for :

Now we can plug in our value into the bottom equation to find our x-value:

So our solution is

We can solve the system using elimination. We can eliminate the x terms by multiplying the bottom equation by :

and now add it to the top equation:

                                      

We plug in our y-value into the bottom equation to get our x-value:

Our solution is then

There are many ways to solve this system of equations. The following is just one way to reach the answer.

Add the two together, to elimnate the y variable. Solve for x and then plug it back in to the first equation to solve for y.

To solve this, let's first try to eliminate x. We can do this by adding the two equations:

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Which implies

We can now solve for x by plugging 2 in for y in either equation.

Thus we have the answer

Let's solve this equation by eliminating the variable x by adding a multiple of the second equation to the first.

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Now let's combine those y values and solve for y.

Now all we have to do is plug that in for y in either original equation to solve for x.

Thus this yields the intersection point

Because one of the variables, z, has already been isolated, let's use the substitution method to solve this system of equations. We know z = 1, so let's plug that into the middle equation to solve for y:

Now that we have found y, let's solve for x by plugging both y and z into the top equation:

Thus we have found that the point of intersection would be

We can solve this system using either substitution or elimination. We'll eliminate them here.

Note: If you wanted to do substitution, we can do it by substituting the top equation into the bottom for .

We'll rearrange the bottom equation to have both y-values aligned and then add the equations:

                                            

Now that we have our x-value, we can find our y-value:

Our answer is then

In order to solve a system of linear equations, we must start by solving one of the equations for a single variable:

We can now substitute this value for y into the other equation and solve for x:

Our last step is to plug this value of x into either equation to find y:

We first move to the left side of the equation:

Subtract the bottom equation from the top one:

Left Side:

Right Side:

So

So dividing by a -1 we get our result.

For any system of linear equations, we can start by solving one equation for one of the variables, and then plug its value into the other equation. In this system, however, we can see that both equations are equal to y, so we can set them equal to each other:

Now we can plug this value for x back into either equation to solve for y:

So the solutions to the system, where the lines intersect, is at the following point:

In order to solve a system of linear equations, we can either solve one equation for one of the variables, and then substitute its value into the other equation, or we can solve both equations for the same variable so that we can set them equal to each other. Let's solve both equations for y so that we can set them equal to each other:

Now we just plug our value for x back into either equation to find y:

So the solution to the system is the point:

Start from equation 3 because it has the least number of variables. We see directly that .

Back substitute into the equation with the next fewest variables, equation 2. Then,

. Solving for , we get

or .

Then back substitute our and into equation 1 to get

.

Solving for x,

.

So our solution to the system is

We can solve the system using elimination. We can eliminate our by multiplying the top equation by :

and then adding it to the bottom equation:

                                          

We can now plug in our y-value into the top equation and solve for our x-value:

Our solution is then