# Algebra 1 : Equations / Solution Sets

## Example Questions

### Example Question #2418 : Algebra 1

Solve the following system of equation

Cannot be solved

Explanation:

Solve for by dividing both sides of the equaion by 6:

Plug this  value into the second equation to solve for :

Subtract 10 from both sides:

Divide by 9:

Plug these and values into the first equation to find :

Combine like terms:

Subtract 2:

Divide by -4:

Therefore the final solution is .

### Example Question #11 : How To Find The Solution For A System Of Equations

Two integers,  and , sum to 16, but when  is doubled, they sum to 34. Find  and .

No solution

Explanation:

When  is doubled to , they sum to 34:

We have two equations and two unknowns, so we can find a solution to this system.

Solve for in the first equation:

Plug this into the second equation:

Solve for :

Use this  value to find . We already have a very simple equation for , .

### Example Question #201 : Algebra

Solve for .

Explanation:

For the second equation, solve for  in terms of .

Plug this value of y into the first equation.

If

and

Solve for  and .

Explanation:

rearranges to

and

, so

### Example Question #1 : Linear Systems With Two Variables

Solve for  in the system of equations:

The system has no solution

Explanation:

In the second equation, you can substitute  for  from the first.

Now, substitute 2 for  in the first equation:

The solution is

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

Without drawing a graph of either equation, find the point where the two lines intersect.

Line 1 :

Line 2 :

Explanation:

To find the point where these two lines intersect, set the equations equal to each other, such that  is substituted with the  side of the second equation. Solving this new equation for  will give the -coordinate of the point of intersection.

Subtract from both sides.

Divide both sides by 2.

Now substitute  into either equation to find the -coordinate of the point of intersection.

With both coordinates, we know the point of intersection is . One can plug in  for  and  for  in both equations to verify that this is correct.

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

What is the sum of and for the following system of equations?

Explanation:

Put the terms together to see that .

Substitute this value into one of the original equaitons and solve for .

Now we know that , thus we can find the sum of and .

### Example Question #21 : How To Find The Solution For A System Of Equations

Two lines have equations of  and . At what point do these lines intersect?

Explanation:

We can solve this problem by setting up a simple system of equations. First, we want to change the equations so one variable can cancel out. Multiplying the first equation by 2 and the second equation by 3 gives us a new system of  and . These equations add up to  or . Plugging in 7 for  in either of the original two equations shows us that  is equal to 1 and the point is .

### Example Question #22 : How To Find The Solution For A System Of Equations

Does this system of equations have one solution, no solutions, or infinite solutions?

one solution:

one solution:

no solution

infinite solutions

infinite solutions

Explanation:

This system has infinite solutions becasue the two equations are actually the exact same line. To discover this, put both equations in terms of y.

First, . Add y to both sides:

Now add 3 to both sides:

Now we can show that the second equation also represents the line

divide both sides by 2

Since both equations are the same line, literally any point on one line will also be on the other - infinite solutions.

### Example Question #23 : How To Find The Solution For A System Of Equations

Find the solution for the system of equations.

and

and

and

and

and

and