Overview
Not all equations can be solved in one step. Many take several steps to solve, but the principles of solving one-step equations can be applied. Use the order of operations to simplify the equation; math properties to combine like terms; then apply inverses to isolate the variable on one side of the equation in order to solve it.
PEMDAS to the Rescue
A messy equation can almost always be cleaned by applying the order of operations. Start simplifying the equation by solving within parentheses, clearing exponents, multiplication, division, addition and subtraction. Sometimes the order of operations will suggest shortcuts. Suppose a problem were (3x –x) +2 =8. Solving within the parentheses, 3x –x =2x + 2 = 8.
Review Math Properties
The commutative properties of addition and multiplication; associative properties of addition and multiplication, and the distributive property of multiplication over addition or subtraction are good strategies when solving equations. Students are able to apply them to tame an equation.
Combine Like Terms
Whenever possible, combine like terms. They become obvious when they are not hiding behind parentheses, standing next to one another in the equation. For example, an expression such as 5x +2 + 3x +4 +6y +3-y can be simplified to 5x + 3x + 6y –y +2 +4 +3 or 8x +5y +9. After like terms are combined, the variable can be isolated on one side of the equation and solved. If 2x +2 = 8, then 2x +2 -2 =8-2 or 2x =6. Both sides can then be divided by 2 so that x = 6/2 or 3.
Multi-Step Equations Represent the Real World
Many problems in the everyday world are modelled by equations that represent more than one step. Suppose that the regular price of a gallon of gasoline is $3.00 a gallon, but a discount card gives 10 cents less per gallon. If a car has an 11 gallon tank, and it is empty, how much will it cost to fill the tank? The first step in solving the problem is to find the cost of the gasoline with the discount card. $3.00 – 0.10 = 2.90 per gallon. The second step is to multiply $2.90 ·11, the number of gallons, or $31.90. In this problem, the x was already isolated on one side of the equation.
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