### All College Algebra Resources

## Example Questions

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

Solve the system of equations.

**Possible Answers:**

None of the other answers are correct.

**Correct answer:**

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 .

**Possible Answers:**

Cannot be determined.

**Correct answer:**

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 #3 : Linear Systems With Two Variables

**What is a solution to this system of equations?**

**Possible Answers:**

**Correct answer:**

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, ?

**Possible Answers:**

More information is needed

**Correct answer:**

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 #5 : 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?

**Possible Answers:**

Nick is 4, Sarah is 8

Nick is 2, Sarah is 6

Nick is 5, Sarah is 15

Nick is 3, Sarah is 9

Nick is 4, Sarah is 12

**Correct answer:**

Nick is 2, Sarah is 6

**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:

**Possible Answers:**

**Correct answer:**

**Solve using elimination:**

multiply the 2nd equation by two to make elimination possible

________________

subtract 2nd equation from the first to solve for

________________

Substitute into either equation to solve for

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

Solve for and :

**Possible Answers:**

**Correct answer:**

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

### Example Question #8 : Linear Systems With Two Variables

If

and

Solve for and .

**Possible Answers:**

None of the available answers

**Correct answer:**

rearranges to

and

, so

### Example Question #9 : Linear Systems With Two Variables

Solve for in the system of equations:

**Possible Answers:**

The system has no solution

**Correct answer:**

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

Now, substitute 2 for in the first equation:

The solution is

### Example Question #10 : Linear Systems With Two Variables

Solve the system of equations.

**Possible Answers:**

**Correct answer:**

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.

Your solution is .