write an equation that expresses the following relationship: six less than twice a number is equal to ten.
Algebraic meaning of the phrases
A single variable represents the unknown number. Let \(x\) denote the number.
| Phrase | Algebraic form (using \(x\)) | Reasoning |
|---|---|---|
| twice a number | \(2x\) | Multiplication by 2 represents doubling. |
| six less than twice a number | \(2x-6\) | “Six less than …” places subtraction after forming the quantity \(2x\). |
| is equal to ten | \(=10\) | Equality connects the expression to a numerical value. |
Equation that expresses the relationship
\[ 2x-6=10. \]
The word order matters: “six less than twice a number” corresponds to \(2x-6\), not \(6-2x\). The subtraction applies to the entire quantity “twice a number.”
Value of the number implied by the equation
Algebraic isolation of \(x\) follows from maintaining equality on both sides.
\[ 2x-6=10 \quad\Longrightarrow\quad 2x=16 \quad\Longrightarrow\quad x=8. \]
Verification by substitution
Substitution of \(x=8\) into the left-hand side reproduces the stated value on the right-hand side.
\[ 2(8)-6=16-6=10. \]
Visualization
Common pitfalls
The phrase “less than” reverses the subtraction relative to the phrase that follows it. Expressions like “six less than twice a number” align with \(2x-6\), while “six less than a number” aligns with \(x-6\).