Solving Basic Algebra Using Steps and Revisions
By Kathleen Cantor, 01 Oct 2020
Algebra is a type of math that focuses on solving expressions that use a combination of symbols and numbers. We call these symbols variables and the numbers constants. For example, x, 3xy, and 6x+1 are all algebraic terms.
Back in the good old days of archaic algebra, schools trained us in the math that develops tools, software. and offers a foundation for many scientific principles. So many things around us use algebra, whether it's the slope of a skating rink, the logistics of a football play, down to how much date night will cost you. It's just that we don't see the math happening behind the scenes!
How it became so notoriously "useless" in our eyes is a mystery. But there's one thing to be said about algebra that justifies students' anxiety surrounding the subject: there's a lot to it.
So many formulae! Algebra is all about finding solutions to everyday problems through expressions. It's a matter of figuring out which problem-solving formula to use and how to recognize them when they show up. If you're looking for extra help, we're going to introduce you to the steps to solving basic algebraic expressions.
You can simply say an expression is an equation that states that a statement on the left-hand side is equal to another on the right-hand side. For example, 2 + 9 = 7 + 4. We know that the sum is true for both sides of the equation. So when we have an equation like this where the statement on the left-hand side is equal to the statement on the right-hand side, we say the equation is a true equation.
By now, you should have a basic idea of what an algebra expression is. It is an equation involving both variables and constants, having distributive operations properties. By this, we mean the operations of addition, subtraction, division, and multiplication.
There are three types of algebraic expressions.
- Monomial: an expression with one unknown variable. Examples of this type of expression are 2x, 5xy.
- Binomial: an expression with two unknown variables. Examples of this type of expression are 2x+8, 4xy+9.
- Trinomial or Polynomial: an expression with three unknown variables. Examples are 8x+57+2z, 2xy+z+7y.
Parts of an Algebra Expression
Let's consider a simple algebraic expression (6x-3).
- Variable: This is usually the letter or symbol that is unknown. In the expression above, it is ''x''.
- Constant: This is usually the number that stands alone without any variable in an algebraic expression. Our constant in the expression is ''3''.
- Coefficient: This is the number multiplied with the variable in an expression. Our coefficient above is ''6'.
- Exponent: Variable ''x'' above has the power of 1, called exponents, or we say x is raised to the power of 1.
Order of Operations for Algebra Expression
We have learned what expressions and equations are. Now let's talk about the basic algebraic equation. We know already that the algebraic equation is an expression of two statements with variables and numbers, where we need to find the variable's value so that the equation becomes a true equation.
We're going to be looking into the types of algebraic equations or expressions and the steps to solving them.
Before we go into these algebraic equations, let us look at these algebraic rules.
- When you have an expression that requires more than one operation to solve for the variable, follow the rules of the Order of Operation.
- If the expression contains parentheses, u multiplication operation on the expression in the bracket first to remove the parenthesis.
- Perform the multiplication and division operation in order from the left-hand to the right-hand side of the equation.
- Perform the addition and subtraction operation in order from the left-hand to the right-hand side of the equation.
- If you have an expression like this: 9 x [(3 x 7) – (10 – 3)], perform the operation starting from the innermost parenthesis first. Multiply the expression in the inner bracket first, then subtract and rewrite the expression.
- Group similar terms to one side of the equation using basic operations, including addition, subtraction, and division.
- Solve for the value of the variable through the operation of multiplication, or division.
The following are operations performed on algebra expressions.
- + = Plus Sign
- - = Minus Sign
- . = Multiplication Sign
- x = Multiplication Sign in scientific notation
- = = Equals
- ≠ does not equal
- ≈ Almost equal
- > = Greater than
- ≥ = Greater than or equal to
- < = Less than
- ≤ = Less than or equal to
- ∫ = Root
Basic Algebra Equations
The Linear Equation
In algebra, the linear equation is an expression having only one equation and one variable to solve. Consider the following equation: 5x+8 = 12.
Applying the rules of Order of Operation, we subtract the constant on the left-hand side of the equation from both sides of the equation. That is 5x+8-8 = 12-8. After performing subtraction, we have 5x = 4. We then perform the operation of division on both sides of the equation. ''X'' will now be 4/5.
You may want to test whether the above expression is a true expression. To do this, you'll substitute the value of ''x'' in the original equation, 5x+8 = 12. That is 5(4/5)+8 = 12. Removing the parenthesis, you get 4+8 = 12, which is a true algebraic equation.
In the example: 9x+6 = 14, we are going to solve for the variable x. Following the rules above, we remove parentheses first. Since there are no parentheses, we move to the next order of operation which is collecting like-terms to one side of the equation.
We move the variable x together with its coefficient to one side of the equation, and also collect numbers to another side of the equation. To do this, we subtract 6 from both sides of the equation.
- 9x+6-6 = 14-6
- 9x = 8.
Finally, we apply the rule which states that solve for the value of the variable through the operation of multiplication or division. We, therefore, divide both sides of the equation by 9. In the end, the solution for variable x is equal to 8/9, i.e = 8/9.
Solved Example of Simultaneous Equation
We will solve for the two variables X and Y in the following equations.
- 2x+6y = 4
- 3x+y = -7
Following the rules again, there are no parentheses, so we'll move to the next rule. To collect like terms to one side of the equation, we take any of the variables and multiply both equations by one of its quotients. Let us take variable Y. Multiply equation 2 by 6, the quotient of variable y for the first equation.
- 2x+6y = 4
- 12x+6y = -42
Perform subtraction operation between the two equations, then subtract equation 1 from 2.
- 10x = 46
Applying the rule again, we divide both side of the equation by 10 to solve for our variable x
- x = 46/10 = 4.6
We see the application of algebra in all spheres of life from manufacturing to medicine, investment, etc. Algebraic formulas can even be useful when you go grocery shopping for your next dinner party. No matter what you're using it for, understanding the steps for solving a basic equation in Algebra can open doors for you to understand more difficult principles.
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