# Algebra 1 : How to find a solution set

## Example Questions

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### Example Question #1 : How To Find A Solution Set

Solve for :

The equation has no solution.

Explanation:

Move one radical to the other side, then square, thereby yielding an equation with only one radical.

Isolate the radical on one side, then square.

Substitution confirms this to be the only solution.

### Example Question #2 : How To Find A Solution Set

If the area of a rectangle is 100 square feet and the width is 20 feet, then what is the perimeter?

50 feet

80 feet

30 feet

20 feet

50 feet

Explanation:

The area of a rectangle is , where A is the area, L is the length, and W is the width.  The perimeter is given by .  We know that  and .  We can solve for L using   The perimeter is then  feet.

### Example Question #3 : How To Find A Solution Set

Solve for :

The equation has no solution.

Explanation:

Cube both sides of the equation to form a linear equation, then solve:

### Example Question #4 : How To Find A Solution Set

Solve this system of equations.

, ,

, ,

, ,

, ,

, ,

, ,

Explanation:

Equation 1:

Equation 2:

Equation 3:

Adding the terms of the first and second equations together will yield .

Then, add that to the third equation so that the y and z terms are eliminated. You will get .

This tells us that x = 1. Plug this x = 1 back into the systems of equations.

Now, we can do the rest of the problem by using the substitution method. We'll take the third equation and use it to solve for y.

Plug this y-equation into the first equation (or second equation; it doesn't matter) to solve for z.

We can use this z value to find y

So the solution set is x = 1, y = 2, and z = –5/3.

### Example Question #4 : How To Find A Solution Set

Solve for

Explanation:

To solve this problem we can first add  to each side of the equation yielding

Then we take the square root of both sides to get

Then we calculate the square root of  which is .

### Example Question #5 : How To Find A Solution Set

Determine where the graphs of the following equations will intersect.

Explanation:

We can solve the system of equations using the substitution method.

Solve for  in the second equation.

Substitute this value of  into the first equation.

Now we can solve for .

Solve for  using the first equation with this new value of .

The solution is the ordered pair .

### Example Question #1 : Solving Equations And Inequallities

Give all real solutions of the following equation:

Explanation:

By substituting  - and, subsequently,  this can be rewritten as a quadratic equation, and solved as such:

We are looking to factor the quadratic expression as , replacing the two question marks with integers with product  and sum 5; these integers are .

Substitute back:

The first factor cannot be factored further. The second factor, however, can itself be factored as the difference of squares:

Set each factor to zero and solve:

Since no real number squared is equal to a negative number, no real solution presents itself here.

The solution set is .

### Example Question #6 : How To Find A Solution Set

Solve for :

Explanation:

Subtract 12 from both sides:

–3x = –15

Divide both sides by –3:

x = 5

### Example Question #1 : How To Find A Solution Set

Solve for :

Explanation:

Distribute the x through the parentheses:

x2 –2x = x2 – 8

Subtract x2 from both sides:

–2x = –8

Divide both sides by –2:

x = 4

### Example Question #3 : How To Find A Solution Set

Solve for :

The equation has no solution.

Explanation:

Add 5 to both sides to isolate the cube root:

Cube both sides:

To isolate , move it to the right side of the equation. We choose the right side instead of the left side so as to make  positive:

Subtract 125 from both sides to isolate :

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