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

Example Questions

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Example Question #33 : Equations / Solution Sets

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 #2 : 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 #276 : Equations / Inequalities

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 #277 : Equations / Inequalities

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 #5 : 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 #2 : How To Find The Solution For A System Of Equations

Find a solution for the following system of equations:

infinitely many solutions

no solution

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 #3 : How To Find The Solution For A System 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 #4 : How To Find The Solution For A System Of Equations

We have three dogs: Joule, Newton, and Toby. Joule is three years older than twice Newton's age. Newton is Tody's age younger than eleven years. Toby is one year younger than Joules age. Find the age of each dog.

Joule: 8 years

Newton: 4 years

Toby: 8 year

Joule: 12 years

Newton: 1 year

Toby: 5 year

none of these

Joule: 9 years

Newton: 3 years

Toby: 8 year

Joule: 5 years

Newton: Not born yet

Toby: 1 year

Joule: 9 years

Newton: 3 years

Toby: 8 year

Explanation:

First, translate the problem into three equations. The statement, "Joule is three years older than twice Newton's age" is mathematically translated as

where  represents Joule's age and  is Newton's age.

The statement, "Newton is Toby's age younger than eleven years" is translated as

where  is Toby's age.

The third statement, "Toby is one year younger than Joule" is

.

So these are our three equations. To figure out the age of these dogs, first I will plug the third equation into the second equation. We get

Plug this equation into the first equation to get

Solve for . Add  to both sides

Divide both sides by 3

So Joules is 9 years old. Plug this value into the third equation to find Toby's age

Toby is 8 years old. Use this value to find Newton's age using the second equation

Now, we have the age of the following dogs:

Joule: 9 years

Newton: 3 years

Toby: 8 years

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