Solving One and Two-Step Equations in Algebra

Study Notes

Algebra involves using letters and symbols to represent numbers and quantities in formulas and equations.
It is a branch of mathematics that deals with generalizing arithmetic operations and solving for unknown values.

One-step equations are algebraic equations that can be solved in only one step.
This involves isolating the variable by performing a single operation (addition, subtraction, multiplication, or division) on both sides of the equation.
For x + 5 = 12, subtract 5 from both sides to get x = 7.
For x - 3 = 8, add 3 to both sides to get x = 11.
For 3x = 15, divide both sides by 3 to get x = 5.
For x / 2 = 6, multiply both sides by 2 to get x = 12.

Two-step equations require two operations to isolate the variable.
These equations often involve a combination of addition or subtraction, followed by multiplication or division (or vice versa).
For 2x + 3 = 9, first subtract 3 from both sides (2x = 6), then divide by 2 (x = 3).
For 4x - 1 = 11, first add 1 to both sides (4x = 12), then divide by 4 (x = 3).
For x / 5 + 2 = 4, first subtract 2 from both sides (x / 5 = 2), then multiply by 5 (x = 10).
For 3x / 2 = 9, first multiply both sides by 2 (3x = 18), then divide by 3 (x = 6).

Substitution involves replacing a variable with its known value or an expression to simplify or solve an equation.
If x = 3, then in the expression 2x + 5, substitute 3 for x: 2(3) + 5 = 6 + 5 = 11.
In a system of equations, solve one equation for one variable and substitute that expression into the other equation to solve for the remaining variable.
Given x + y = 7 and x = 2, substitute 2 for x in the first equation: 2 + y = 7, so y = 5.

The distributive property states that a(b + c) = ab + ac.
It allows you to multiply a single term by two or more terms inside a set of parentheses.
For example, 3(x + 2) = 3x + 6.
Distribution is also applicable to subtraction: a(b - c) = ab - ac.
5(x - 4) = 5x - 20.
The distributive property can be combined with simplifying expressions to solve equations.
2(x + 3) + 4 = 16 simplifies to 2x + 6 + 4 = 16, then 2x + 10 = 16, and finally 2x = 6, so x = 3.

Decimals are numbers expressed in base 10 notation, containing a whole number part and a fractional part separated by a decimal point.
Algebraic equations can include decimal numbers as coefficients, constants, or solutions.
Use the same algebraic principles to solve equations with decimals as you would with integers or fractions.
For example, to solve 0.2x + 1.5 = 2.5, subtract 1.5 from both sides (0.2x = 1), then divide by 0.2 (x = 5).
When dealing with multiple decimal terms, it may be helpful to clear the decimals by multiplying all terms by a power of 10 to simplify calculations.
For instance, in the equation 0.1x + 0.05 = 0.2, multiply all terms by 100: 10x + 5 = 20, then 10x = 15, so x = 1.5.

Simplifying expressions involves reducing an algebraic expression to its simplest form by combining like terms and performing operations.
Combine like terms (terms with the same variable raised to the same power).
For example, 3x + 2x - x simplifies to 4x.
Apply the order of operations (PEMDAS/BODMAS): Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction.
2(x + 3) - 4 simplifies to 2x + 6 - 4, then 2x + 2.
Use the distributive property to remove parentheses.
Apply the commutative property to rearrange terms (e.g., a + b = b + a).
Apply the associative property to regroup terms (e.g., (a + b) + c = a + (b + c)).
Simplifying expressions can make equations easier to solve.
Starting from 4x + 2(x - 1) = 10, distribute to get 4x + 2x - 2 = 10, combine like terms to get 6x - 2 = 10, add 2 to both sides to get 6x = 12, and divide by 6 to get x = 2.