# Algebra 1 : Equations / Solution Sets

## Example Questions

### Example Question #1 : 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 #5 : 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 :

### Example Question #1 : Setting Up Expressions

We have three cats, Chai, Sora, and Newton. Chai is 3 years old. Sora two years older than twice Chai's age. Newton is one year younger than one-fourth of Sora's age. How old are Sora and Newton?

Sora: 3 years

Newton: not born yet

Sora: 4 years

Newton: one half year

Sora: 1 year

Newton: 8 years

Sora: 8 years

Newton: 1 year

Sora: 5 years

Newton: 6 years

Sora: 8 years

Newton: 1 year

Explanation:

To make this much easier, translate the word problem into a system of three equations.

We have C for Chai, S for Sora, and N for Newton. To find Sora's age, plug in  into .

Sora is 8 years old. Use this to find Newton's age.

Newton is one year old.  So the answer is:

Sora, 8 years

Newton, 1 year

### Example Question #1 : Solutions And Solution Sets

Give all real solutions of the following equation:

The equation has no real solutions.

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 36 and sum ; these integers are .

Substitute back:

These factors can themselves be factored as the difference of squares:

Set each factor to zero and solve:

The solution set is .

### Example Question #673 : Intermediate Single Variable Algebra

Solve for :

There is no solution.

Explanation:

Subtract 1 from both sides, then multiply all sides by :

A quadratic equation is yielded. We can factor the expression, then set each individual factor to 0.

Both of these solutions can be confirmed by substitution.

### Example Question #11 : How To Find A Solution Set

Solve for :

Explanation:

Isolate the radical, square both sides, and solve the resulting quadratic equation.

Factor the expression at left by finding two integers whose product is 65 and whose sum is ; they are .

Set each linear binomial to 0 and solve for  to find the possible solutions.

or

Substitute each for .

This is a false statement, so 5 is a false "solution".

This is a true statement, so 13 is the only solution of the equation.

### Example Question #12 : How To Find A Solution Set

Solve the equation for .

Explanation:

The two sides of the equation will be equal if the quantities inside the absolute value signs are equal or equal with opposite signs.

From this, you get two equations.

or

### Example Question #13 : How To Find A Solution Set

Find the solution, in the form , to the following system of equations:

Explanation:

Multiply (1) by 3, multiply (2) by 4:

Substitute  into (1) and solve for :