# Algebra 1 : How to find the solution for a system of equations

## Example Questions

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### Example Question #1 : How To Find The Solution For A System Of Equations

A cube has a volume of  . If its width is  , its length is , and its height is  , find .

Explanation:

Since the object in question is a cube, each of its sides must be the same length. Therefore, to get a volume of  , each side must be equal to the cube root of , which is  cm.

We can then set each expression equal to .

The first expression   can be solved by either  or , but the other two expressions make it evident that the solution is .

### Example Question #1 : How To Find The Solution For A System Of Equations

Solve the system for  and .

Explanation:

The most simple method for solving systems of equations is to transform one of the equations so it allows for the canceling out of a variable. In this case, we can multiply  by  to get .

Then, we can add to this equation to yield , so .

We can plug that value into either of the original equations; for example, .

So,  as well.

### Example Question #1 : How To Find The Solution For A System Of Equations

What is the solution to the following system of equations:

Explanation:

By solving one equation for , and replacing  in the other equation with that expression, you generate an equation of only 1 variable which can be readily solved.

### Example Question #1 : How To Find The Solution For A System Of Equations

Solve this system of equations for :

None of the other choices are correct.

Explanation:

Multiply the bottom equation by 5, then add to the top equation:

### Example Question #2 : How To Find The Solution For A System Of Equations

Solve this system of equations for :

None of the other choices are correct.

Explanation:

Multiply the top equation by :

### Example Question #1 : How To Find The Solution For A System Of Equations

Solve this system of equations for :

None of the other choices are correct.

Explanation:

Multiply the top equation by :

### Example Question #1 : How To Find The Solution For A System Of Equations

Find the solution to the following system of equations.

Explanation:

To solve this system of equations, use substitution. First, convert the second equation to isolate .

Then, substitute  into the first equation for .

Combine terms and solve for .

Now that we know the value of , we can solve for using our previous substitution equation.

### Example Question #1 : How To Find The Solution For A System Of Equations

Find a solution for the following system of equations:

no solution

infinitely many solutions

no solution

Explanation:

When we add the two equations, the  and  variables cancel leaving us with:

which means there is no solution for this system.

### Example Question #151 : Systems Of Equations

Solve for :

Explanation:

First, combine like terms to get . Then, subtract 12 and from both sides to separate the integers from the 's to get . Finally, divide both sides by 3 to get .

### Example Question #151 : Systems Of Equations

We have two linear functions:

Find the coordinate at which they intersect.

none of these

Explanation:

We are given the following system of equations:

We are to find  and . We can solve this through the substitution method.  First, substitute the second equation into the first equation to get

Solve for  by adding 4x to both sides