Writing an equation is the bridge between a situation and its solution. The goal is not to grab every number and combine it somehow; it is to name the unknown, preserve each stated relationship, and connect two equal quantities with an equal sign.
Learning objectives
- Distinguish an equation from an expression and interpret the equal sign as a balance.
- Define variables with quantities and units before translating a relationship.
- Translate operation and equality phrases without reversing order or losing grouping.
- Model consecutive integers and consecutive even or odd integers correctly.
- Check whether an equation matches a situation without necessarily solving it.
What an equation says
- Equation
- A mathematical statement that two expressions have the same value. The equal sign in \(4x+3=27\) separates the two expressions and asserts that they are equal.
Expression versus equation
Expression
\(5n-8\) names a quantity. It can be simplified or evaluated, but it does not claim equality and cannot be solved by itself.
Equation
\(5n-8=42\) states that two quantities match. It can be tested for truth and solved for values that make it true.
A reliable translation process
Unknown → relationships → equality → audit
- Identify the unknown
Ask exactly what quantity is not specified. Include units when useful: let \(t\) be the travel time in hours.
- Define a variable
Choose a symbol and state its meaning. If other unknown-looking quantities are described in relation to it, express them using the same variable.
- Translate each relationship
Build grouped phrases from the inside out. Preserve the named order for subtraction and division.
- Locate equality language
Words such as ‘is,’ ‘equals,’ ‘is identical to,’ and ‘is the same as’ identify the two equal sides.
- Audit the model
Read the equation back in words, check units, and test a simple value when order or grouping is uncertain.
Translate operation and equality language
| Language | Meaning | Original translation |
|---|---|---|
| is; equals; is identical to; is the same as | equality | ‘\(3\) more than \(x\) is \(20\)’ → \(x+3=20\) |
| sum; more than; increased by | addition | ‘\(p\) increased by \(11\)’ → \(p+11\) |
| difference of; minus; decreased by | subtraction in named order | ‘the difference of \(a\) and \(7\)’ → \(a-7\) |
| less than | subtraction with reversed spoken order | ‘\(6\) less than \(m\)’ → \(m-6\) |
| product; times; of | multiplication | ‘three-fourths of \(q\)’ → \(\frac34q\) |
| quotient; divided by; per | division in named order | ‘the quotient of \(r\) and \(5\)’ → \(\frac r5\) |
Grouping preserves the named quantity
An outside operation must apply to the entire phrase it modifies. ‘Twice the sum of \(x\) and \(7\)’ means form the sum first, then double it: \(2(x+7)\). Writing \(2x+7\) doubles only \(x\), so it represents a different relationship.
Translate a grouped fractional relationship
One-third of the difference between a number and \(8\) is the same as \(5\). Write an equation.
- Define
Let \(n\) represent the unknown number.
- Build the difference
‘The difference between a number and \(8\)’ is \(n-8\).
- Apply the fraction
One-third of that complete difference is \(\frac{n-8}{3}\).
- Create equality
‘Is the same as \(5\)’ supplies the equal sign and right side.
Consecutive-number models
Consecutive values follow a fixed step. Ordinary consecutive integers differ by \(1\). Consecutive even integers and consecutive odd integers each differ by \(2\). The algebraic pattern is the same for even and odd lists; the starting variable must have the required parity.
| Sequence type | First four terms | Required starting value |
|---|---|---|
| Consecutive integers | \(n,\ n+1,\ n+2,\ n+3\) | \(n\) is any integer |
| Consecutive even integers | \(n,\ n+2,\ n+4,\ n+6\) | \(n\) is even |
| Consecutive odd integers | \(n,\ n+2,\ n+4,\ n+6\) | \(n\) is odd |
Model consecutive odd integers
The sum of three consecutive odd integers is \(189\). Write an equation and identify what each term represents.
- Define
Let \(n\) be the least odd integer.
- Represent the sequence
The next two odd integers are \(n+2\) and \(n+4\).
- Translate the sum
Add the three representations: \(n+(n+2)+(n+4)\).
- State equality
The word ‘is’ connects that sum to \(189\).
Equations with two variables
An equation may describe a relationship without determining a unique numerical value. If adult tickets cost \(a\) dollars and student tickets cost \(s\) dollars, ‘an adult ticket costs \(4\) dollars more than a student ticket’ becomes \(a=s+4\). Both variables are meaningful even though no price can yet be calculated.
Model two related quantities
A rectangular garden has length \(L\) meters. Its width is \(3\) meters less than half its length. Write an equation relating width \(W\) and length \(L\).
- Identify quantities
The two defined measurements are \(W\) and \(L\).
- Translate half
Half the length is \(\frac L2\).
- Translate less than
Three less than that quantity is \(\frac L2-3\).
- State the relationship
The width equals the translated expression.
Common mistakes and SAT checks
Check your understanding
Which equation represents ‘three times the sum of \(x\) and \(4\) is \(27\)’?
- \(3x+4=27\)
- \(3(x+4)=27\)
- \(x+12=27\)
- \(3x+4x=27\)
Show answer and explanation
Answer: \(3(x+4)=27\)
The complete sum \(x+4\) must be formed before it is multiplied by \(3\). The word ‘is’ supplies the equal sign.
Key takeaways
What to remember
- Define the unknown quantity and its units before writing symbols.
- Treat the equal sign as a statement that two complete expressions have the same value.
- Preserve order in subtraction and division, and treat ‘less than’ as a reversal warning.
- Use parentheses or a fraction bar when an operation applies to a multi-term quantity.
- Represent consecutive integers with steps of \(1\), and consecutive even or odd integers with steps of \(2\) plus the appropriate parity condition.
- Read the equation back in words and use a simple test value to audit order and grouping.
Put these notes into practice
Apply the ideas with SAT-style questions, then reinforce key details with flashcards.