Algebra Test and System of Equations

Algebra test, either in your general examinations or specifically designed to measure your aptitude, requires you to know various equations.

What is an equation?

An equation is an open algebraic expression which involves a sign of equality (=). For example, x+7 = 49 is an equation. There are various types of algebra equations. A few of them are frequently used like: 

1-    Linear Equation

2-    Quadratic Equations

3-    Radical equations

4-    Two simultaneous Equations

Linear Equations

The linear equation involves one unknown and in which the exponent of the unknown is no larger than 1. For example, 4x= 16, is a linear equation. You should watch some precautions while solving linear equations in an algebra test.

1-    Remove fractions and decimals by multiplication.

2-    You should collect all unknown variables for which you are solving on the same side of the equation. When you see a term crossing the equal sign from one side of the equation to the other, must change the sign.

3-    Combine similar terms to determine the coefficient of the unknown variable. When terms cannot be combined, then go for the combination of factoring.

4-    Then divide both sides of the equations with the coefficient you get.

5-    When you are going to solve an equation with two unknowns, you will have to work with two equations, simultaneously. You are required to eliminate one of the two unknown variables and then solve the resulting single unknown equation.

Factorization

What are factors in equation x²+xy?

They are x and x+y...

...because their product x(x + y) is equal to x²+xy.

Hence factors of an expression are two or more or more expressions whose product is equal to the first expression. Please note that to factor any expression is to write the expression as the product of its factors. There are following types of factoring:

Common Factor

A common factor can divides into each term of the expression. When you want to factor an expression with a common factor in an algebra test, you need to find the largest common factor of all the terms in the expression. Then divide the expression by the largest common factor. And finally write the answer as product of the two factors.

Difference of Two Squares

An expression having two terms and consisting of one perfect square minus another perfect square, the expression is called a difference of two squares. For example, a²-b² is difference of two squares.

To factor a difference of two squares in any algebra test, you need to take the square root of each of the perfect squares. One of the factors is the sum of the square roots. The other factor is the difference of the square root. You will find the product of the two factors.

Factoring Trinomials into Binomials

A trinomial is an expression with 3 unlike terms such as a²+2a+1. A binomial is an expression with 2 unlike terms such as x+2 or a+b. When you are required to factor trinomials into binomials, you should beware of the catch that all trinomials can not be factored into binomials. And there is no quick way to know this catch unless you attempt to factor.

Quadratic Equation

A quadratic equation is always in the form of ax² + bx + c = 0 in which a, b, c are rational numbers and a is not equal to 0. It is a standard form of the quadratic equation in any algebra test.

Radical Equation

Radical equation contains the variables under a radical sign. When you are required to solve a radical equation in algebra tests, you should always get the radical alone on one side of the equation and then square both sides to remove the radical to solve. You should always check solutions to the radical equations because squaring the both sides may sometimes result into extraneous roots.

For example:

√x+5 = 49
x = 44

Check: √49 equals to 7 so the solution is correct.

Two Simultaneous Equations

When you have two distinct equations, you can solve for the two variables. Two forms of the same equation are not sufficient. You should combine the equations in a way that one variable cancels out the other.

How to Solve Equations in an Algebra Test

While solving equations in any algebra test, you need to have not only understanding but also a lot of practice. As a last minute reminder, please keep the following important tips in your mind.

1- Read each algebraic question carefully. You shall find different relations between the quantities given in the problem.

2- Give symbols to unknown quantities such as x, y, z.

3- Redefine the conditions mentioned in symbolic language. You shall obtain an equation.

4- Then solve the equation obtained to known the unknown quantities.

5- Check your results.

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