# College Algebra : Linear Systems with Two Variables

## Example Questions

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

Solve the system of equations.

None of the other answers are correct.

Explanation:

Isolate  in the first equation.

Plug into the second equation to solve for .

Plug into the first equation to solve for .

Now we have both the and values and can express them as a point: .

### Example Question #2 : Linear Systems With Two Variables

Solve for  and .

Cannot be determined.

Explanation:

1st equation:

2nd equation:

Subtract the 2nd equation from the 1st equation to eliminate the "2y" from both equations and get an answer for x:

Plug the value of  into either equation and solve for :

### Example Question #1 : Solving Equations

What is a solution to this system of equations?

Explanation:

Substitute equation 2. into equation 1.,

so,

Substitute  into equation 2:

so, the solution is .

### Example Question #4 : Linear Systems With Two Variables

A man in a canoe travels upstream 400 meters in 2 hours. In the same canoe, that man travels downstream 600 meters in 2 hours.

What is the speed of the current, , and what is the speed of the boat in still water, ?

Explanation:

This problem is a system of equations, and uses the equation .

Start by assigning variables. Let  stand for the rate of the boat, let  stand for the rate of the current.

When the boat is going upstream, the total rate is equal to .  You must subtract because the rates are working against each other—the boat is going slower than it would because it has to work against the current.

Using our upstream distance (400m) and time (2hr) from the question, we can set up our rate equation:

When the boat is going downstream, the total rate is equal to  because the boat and current are working with each other, causing the boat to travel faster.

We can refer to the downstream distance (600m) and time (2hr) to set up the second equation:

From here, use elimination to solve for  and .

1. Set up the system of equations, and solve for .

2. Subsitute  into one of the equations to solve for .

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

Nick’s sister Sarah is three times as old as him, and in two years will be twice as old as he is then.  How old are they now?

Nick is 2, Sarah is 6

Nick is 4, Sarah is 12

Nick is 5, Sarah is 15

Nick is 4, Sarah is 8

Nick is 3, Sarah is 9

Nick is 2, Sarah is 6

Explanation:

Step 1: Set up the equations

Let = Nick's age now
Let = Sarah's age now

The first part of the question says "Nick's sister is three times as old as him".  This means:

The second part of the equation says "in two years, she will be twice as old as he is then).  This means:

Add 2 to each of the variables because each of them will be two years older than they are now.

Step 2: Solve the system of equations using substitution

Substitute for in the second equation.  Solve for

Plug into the first equation to find

### Example Question #6 : Linear Systems With Two Variables

Solve the system of equations:

Explanation:

Solve using elimination:

multiply the 2nd equation by two to make elimination possible
________________

subtract 2nd equation from the first to solve for
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Substitute  into either equation to solve for

### Example Question #7 : Linear Systems With Two Variables

Solve for  and :

Explanation:

There are two ways to solve this:

-The 1st equation can be mutliplied by  while the 2nd equation can be multiplied by  and added to the 1st equation to make it a single variable equation where

.

This can be plugged into either equation to get

or

-The 2nd equation can be simplified to,

.

This value for  can then be substituted into the first equation to make the equation single variable in .

Solving, gives , which can be plugged into either original equation to get

If

and

Solve for  and .

Explanation:

rearranges to

and

, so

### Example Question #1 : How To Find The Solution For A System 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 #9 : Linear Systems With Two Variables

Solve the system of equations.

Explanation:

Use elimination, multiply the top equation by -4 so that you can eliminate the X's.

__________________

Combine these two equations, and then you have;

Plug in y into one of the original equations and solve for x.