Precalculus : Systems of Equations

Example Questions

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Example Question #11 : Inequalities And Linear Programming

Find the point of intersection by using Gaussian elimination:

Explanation:

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.

Example Question #12 : Inequalities And Linear Programming

Solve the following system of equations:

Explanation:

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

Example Question #11 : Inequalities And Linear Programming

Solve the following system of equations for the intersection point in space:

Explanation:

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

Example Question #1 : Solve Nonlinear Systems Of Equations

Solve the following system of nonlinear equations:

Explanation:

We can start by rearranging each equation so that y is on one side:

Now that both equations are equal to y, we can set the right sides equal to each other and solve for x:

Our last step is to plug these values of x into either equation to find the y values of our solutions:

So our solutions are the folloiwing two points:

Example Question #1 : Solve Nonlinear Systems Of Equations

Solve the following system of nonlinear equations:

Explanation:

Our first step is to rearrange each equation so that the left side is just y:

Now that both equations are equal to y, we can see that the right sides of each equation are equal to each other, so we set this up below and solve for x:

Our last step is to plug these values of x into either equation to solve for the y values of our solutions:

So the solutions to the system are the following points:

Example Question #3 : Solve Nonlinear Systems Of Equations

Solve the following system:

Explanation:

Our first step is to solve the bottom equation for

We can now substitute it into the top equation:

and solve for our  values:

our values are then:

We can now plug our values into the bottom equation we had solved for  and arrive at our solutions:

So, our solutions are

Example Question #4 : Solve Nonlinear Systems Of Equations

Solve the following system:

Explanation:

Our first step is to solve the bottom equation for

so we can substitute it into the top equation:

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

Remember we cannot take a square root of a negative number without getting an imaginary number. As such, we'll just focus on the  values.

Our solution is then:

Example Question #1 : Solve Nonlinear Systems Of Equations

Solve the following system for :

Explanation:

The first step is to solve the bottom equation for

since our question specifies for  we just focus on

We now substitute this equation into the top equation:

we can now plug in our x-values into the bottom equation to find our y-values:

The solutions are then:

Example Question #1 : Solve Nonlinear Systems Of Equations

Solve the following system:

Explanation:

We can solve this equation by using substitution since the bottom equation is already solved for . Substituting the bottom equation into the top we get:

We then solve the equation for our  values:

Finally, we substitute our values into the bottom equation to get our  values:

Our different solutions are then:

Example Question #7 : Solve Nonlinear Systems Of Equations

Solve the following system:

Explanation:

We can substitute the top equation into the bottom:

and solve for  values:

Now that we have our  values we can plug it into the top equation and find our  values

So, our values are

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