# Precalculus : Inequalities and Linear Programming

## Example Questions

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### Example Question #1 : Inequalities And Linear Programming

Solve the following system of linear equations:

Explanation:

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:

### Example Question #1 : Solve Systems Of Linear Equations

Solve the system of linear equations for :

Explanation:

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.

### Example Question #1 : Systems Of Equations

Solve the following system of linear equations:

Explanation:

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:

### Example Question #1 : Inequalities And Linear Programming

Solve the following system of linear equations:

Explanation:

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:

### Example Question #5 : Inequalities And Linear Programming

Use back substitution to solve the system of linear equations.

Explanation:

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

### Example Question #6 : Inequalities And Linear Programming

Solve the following system:

Explanation:

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

### Example Question #7 : Inequalities And Linear Programming

Solve the following system:

Explanation:

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

### Example Question #1 : Solve Systems Of Linear Equations

Solve the following system:

Explanation:

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

### Example Question #9 : Inequalities And Linear Programming

Solve the following system:

Explanation:

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:

### Example Question #10 : Inequalities And Linear Programming

Solve the following system of equations: