Solving One-Step Equations
Students solve and check one-step equations involving nonnegative rational numbers using inverse operations.
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Understand Equality
An equation states that two expressions have the same value. The equal sign means “is equal to,” not “the answer comes next.” Think of an equation as a balanced scale: both sides must have equal weight. For example, x + 3 = 8 says that an unknown number plus 3 has the same value as 8. The value x = 5 makes the equation true because 5 + 3 = 8. A solution is a value that makes both sides equal. If the same number is added to, subtracted from, multiplied by, or divided out of both sides, the equation stays balanced, as long as division is not by zero.
Model Equations
A model helps show how the quantities in an equation are related. For x + 2.5 = 6, draw a tape with a total length of 6 units. Divide it into one unknown part labeled x and one part with length 2.5. The missing length is 3.5 because 3.5 + 2.5 = 6. A multiplication equation can also be modeled. In 3x = 9, three equal sections together have a total length of 9. Each section must have length 3, so x = 3. Models make it easier to see whether quantities are being joined, separated, or arranged in equal groups before choosing an operation.
Use Inverse Operations
Inverse operations undo each other. Addition and subtraction are inverse operations, and multiplication and division are inverse operations. To isolate a variable, use the inverse of the operation performed on it. Whatever operation you use must be applied to both sides so equality is preserved. In x + 1.4 = 5, subtract 1.4 from both sides: x + 1.4 − 1.4 = 5 − 1.4, so x = 3.6. In 2.5x = 10, divide both sides by 2.5: 2.5x ÷ 2.5 = 10 ÷ 2.5, so x = 4. Division isolates x when its coefficient is greater than zero.
Solve Addition Equations
To solve an equation of the form x + p = q, subtract p from both sides. This undoes the addition and leaves x alone. Consider x + 2/3 = 1 1/4. Subtract 2/3 from each side: x = 1 1/4 − 2/3. Rewrite 1 1/4 as 5/4, and use a common denominator: 5/4 = 15/12 and 2/3 = 8/12. Then x = 15/12 − 8/12 = 7/12. The solution is x = 7/12. Writing the same subtraction on both sides is important because it keeps the equation balanced. Estimate first when helpful: the answer should be a little more than one-half.
Solve Multiplication Equations
To solve an equation of the form px = q, divide both sides by the coefficient p when p is greater than zero. The coefficient is the number multiplying the variable. For example, solve 3/4x = 6. Divide both sides by 3/4: x = 6 ÷ 3/4. Dividing by a fraction is the same as multiplying by its reciprocal, so x = 6 × 4/3 = 8. Therefore, x = 8. You can also reason that three-fourths of 8 is 6. Applying division to both sides keeps their values equal and isolates x. The solution is nonnegative, as required for the equations in this lesson.
Check Solutions
Check a solution by substituting it for the variable in the original equation. Then simplify both sides. If the two sides have the same value, the solution is correct. Suppose the equation is 1.5x = 4.5 and you found x = 3. Substitute 3 for x: 1.5(3) = 4.5. Since 4.5 = 4.5, the solution checks. For an addition example, if x + 0.8 = 2.1 and x = 1.3, substitute to get 1.3 + 0.8 = 2.1. This simplifies to 2.1 = 2.1. Checking can catch calculation errors, incorrect inverse operations, or a value copied incorrectly.