What are Algebraic Equations?
Equations are statements in mathematics that demonstrate two expressions being equal. An ‘equal to’ (=) symbol is used in equations to link two expressions. Equations consisting of constants, variables, coefficients along with arithmetic operations are called algebraic equations. We will now focus on linear equations with only one variable.
The solution of any algebraic equation is a value of the variable that satisfies the equation. These solutions can be obtained using addition, subtraction, multiplication and division operations.
What are Two-Step Equations?
Equations that can be solved in precisely two steps are called two-step equations in algebra. Two-step equations are represented in the form, Px + Q = R.
Where,
P is the coefficient of the variable, x
Q and R are the constants.
Examples of such equations are:
- 3x + 5 = 7
- 0.5 y + 2 = 1
- \(\frac{1}{3}\) x – 1 = 7
A two-step equation can be solved using:
- Addition Property of Equality
- Subtraction Property of Equality
- Multiplication Property of Equality
- Division Property of Equality
[Note: If there are more than one variable terms in the equation then combine the like terms and solve.]
Steps for solving Two-Step Equations
Step 1: Write the given equation
Step 2: Combine like terms
Step 3: Apply the property of equality such that the variable is on one side and the constant on the other side of the equation
Step 4: Simplify.
Learn More About Two-Step Equations in this Video
Solved Examples
Example 1: Solve 4x – 6 = 18.
Solution:
4x – 6 = 18 [Write the equation]
4x – 6 + 6 = 18 + 6 [Addition property of equality]
4x = 24 [Simplify]
\(\frac{\text{4x}}{4}=\frac{24}{4}\) [Division Property of Equality]
x = 6 [Simplify]
So, the solution of 4x – 6 = 18 is x = 6.
Example 2: Solve the equation 48x + 2x + 10 = 160 and check your solution.
Solution:
48x + 2x + 10 = 160 [Write the equation]
50x + 10 = 160 [Combine like terms]
50x + 10 – 10 = 160 – 10 [Subtraction property of equality]
50x = 150 [Simplify]
\(\frac{\text{50x}}{50}=\frac{150}{50}\) [Division Property of Equality]
x = 3 [Simplify]
So, the solution of 48x + 2x + 10 = 160 is x = 3.
Substitute x = 3 in the equation and check whether the equation is satisfied or not:
48 × 3 + 2 × 3 + 10 = 160
144 + 6 + 10 = 160
160 = 160
LHS = RHS
Hence, the equation is satisfied for x = 3.
Example 3: You wish to decorate a rectangular table mat. You paste a 120 meter frill along the boundary of the mat. What is the length of the mat if its width is 20 meters?
Solution:
P = 2l + 2w [Perimeter of a rectangle]
120 = 2l + 2(20) [Substitute for P and w]
120 = 2l + 40 [Multiply]
120 – 40 = 2l + 40 – 40 [Subtraction Property of Equality]
80 = 2l [Simplify]
\(\frac{80}{2}=\frac{\text{2l}}{2}\) [Division Property of Equality]
40 = l [Simplify]
or, l = 40m
Hence, the length of the table mat is 40 meters.
An equation with the highest exponent of a variable equal to 1 is called a linear equation.
The addition property of equality states that adding the same number to both sides of an equation keeps both sides of the equation equal.
The division property of equality states that dividing both sides of an equation by the same number keeps both sides of the equation equal.
Linear equations are algebraic equations that have the highest exponent of variables as 1, and nonlinear equations are algebraic equations that have the exponent of variables as 2 or more than 2.