Lesson 29: What is an Equation?

§Text Passage

1Summary 重點摘要

Equation is a number sentence with unknown. Use x for unknown, solving means find x. Learn to recognise and write equations.

2Vocabulary cards Vocabulary

3Step by step Steps

1. Block 1: What is an Equality?An equality is a number sentence with an equals sign showing both sides are equal. For example 3 + 4 = 7: the value of 3 + 4 on the left matches the 7 on the right, so it is an equality. Why stress the balance? Because the equals sign declares equilibrium: the values on both sides must be exactly the same, the founding idea of all algebra. Any sentence with an equals sign and matching values counts as an equality; a wrong one, such as 3 + 4 = 8, is no equality at all. A common error is reading the equals sign as 'the answer is', producing chain mistakes like 3 + 4 = 7 = 7 + 2, where a middle step is not equal at all.
2. Block 2: What is an Equation?An equation is an equality containing an unknown. For example x + 3 = 7, where x is the unknown, standing for a number we do not yet know and must find. An equality plus an unknown makes an equation; an equality without an unknown is only an ordinary equality, and the two must be kept apart. Why introduce an unknown? Because many questions run backwards: the result is known and the condition sought, and the unknown holds the wanted number's place so the relation can be written. A common error is calling any lettered expression an equation; two conditions must hold together — an equals sign and an unknown, with neither absent.
3. Block 3: How to Write?To turn a word problem into an equation, first name the unknown x. For example, the problem says 'a number minus 4 equals 9': use x for the number, write 'minus 4' as - 4 and 'equals' as =, giving the equation x - 4 = 9. Why is naming the unknown the first step? Because until the wanted number has a name, no relation can be written; once x is set, every quantity word in the text is translated into a mathematical symbol in turn, and the problem becomes a sentence of its own. A common error is mistranslating the relation, such as writing '4 less than a number' as x + 4; after translating, read the equation once and confirm it matches the wording before moving on.
4. Block 4: How to Solve?The goal of solving is to leave x standing alone on one side of the equals sign. For example x + 3 = 7: subtract 3 from both sides, the left becomes x, the right becomes 7 - 3 = 4, so x = 4. Why subtract from both sides? Because the two sides were balanced, and doing the same thing to both keeps the balance while removing the +3 beside x. Check: substitute 4 back, 4 + 3 = 7, matching the original equation, so the solution is right. A common error is operating on one side only, subtracting 3 on the left while the right stays untouched, which breaks the balance at once; every operation must happen on both sides, the iron rule of solving.
5. Block 5: Balance IdeaAn equation behaves like a balance scale with the equals sign as the beam and the two sides as pans. Adding, subtracting, multiplying or dividing the same number on both sides (division by 0 excepted) keeps the scale level, so the equality still holds. This 'do the same to both sides' principle is the basis of every solving method. Why is division by 0 excepted? Because dividing by 0 is meaningless; no number may be divided by 0. For example x ÷ 2 = 6: multiply both sides by 2 and x = 12, the pans level throughout and the answer reliable. A common error is doing different operations on the two sides, multiplying one and subtracting the other; picturing the scale exposes the tilt at once.
6. Block 6: Common TypesFour equation types are common in primary school: x + a = b, solved by subtracting a on both sides; x - a = b, solved by adding a; ax = b, solved by dividing by a; and x ÷ a = b, solved by multiplying by a. Each uses the opposite operation to clear the number beside x: addition cleared by subtraction, subtraction by addition, multiplication by division, division by multiplication. With practice the approach is obvious at a glance: for 4x = 12, divide both sides by 4 and x = 3. A common error is picking the wrong opposite operation, subtracting 5 again in x - 5 = 9 to get x = 4; first see which operation sits beside x, apply the opposite to both sides, and finish with a check.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Equation = number sentence with unknown
• Unknown usually x
• Solve = find x
• Same operation both sides, still equal
• Always check