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

## Example Questions

### Example Question #21 : 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 #2421 : Algebra 1

Read, but do not solve, the following problem:

Adult tickets to the zoo sell for $11; child tickets sell for$7. One day, 6,035 tickets were sold, resulting in $50,713 being raised. How many adult and child tickets were sold? If and stand for the number of adult and child tickets, respectively, which of the following systems of equations can be used to answer this question? Possible Answers: Correct answer: Explanation: 6,035 total tickets were sold, and the total number of tickets is the sum of the adult and child tickets, . Therefore, we can say . The amount of money raised from adult tickets is$11 per ticket mutiplied by  tickets, or  dollars; similarly,  dollars are raised from child tickets. Add these together to get the total amount of money raised:

These two equations form our system of equations.

### Example Question #2 : Inequalities

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 #2 : Systems Of Equations

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 #6 : 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 #1 : 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 #22 : 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 #23 : How To Find The Solution For A System Of Equations

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

infinite solutions

one solution:

one solution:

no solution

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 #2431 : Algebra 1

A blue train leaves San Francisco at 8AM going 80 miles per hour. At the same time, a green train leaves Los Angeles, 380 miles away, going 60 miles per hour. Assuming that they are headed towards each other, when will they meet, and about how far away will they be from San Francisco?

The two trains will never meet.

Around 2:45AM, about 200.15 miles away from San Francisco

Around 3AM the next day, about 1,520 miles away from San Francisco

Around 10:43AM, about 217.12 miles away from San Francisco

Around 10:43AM, about 217.12 miles away from San Francisco

Explanation:

This system can be solved a variety of ways, including graphing. To solve algebraically, write an equation for each of the different trains. We will use y to represent the distance from San Francisco, and x to represent the time since 8AM.

The blue train travels 80 miles per hour, so it adds 80 to the distance from San Francisco every hour. Algebraically, this can be written as .

The green train starts 380 miles away from San Francisco and subtracts distance every hour. This equation should be .

To figure out where these trains' paths will intersect, we can set both right sides equal to each other, since the left side of each is y.