Collect matching variable terms and substitute with complete brackets.
Core & Extended · Syllabus C2.1, C2.2, E2.1, E2.2
Notes and key methods
An expression represents a value; an equation states that two values are equal. Like terms have the same variables raised to the same powers. Their coefficients can be added, while unlike terms remain separate.
Substitute into the entire expression, including each negative sign and power. A verbal description such as “three less than twice n” gives 2n−3, whereas “twice three less than n” needs its grouping clarified. Define the quantity represented by each letter before constructing a model.
Check the conditions
Use the stated units and precision.
Show the method as well as the final answer.
Three worked examples
Example 1 · Expressions, substitution and like terms
Consider \(E=6 x + 4 y + 28\).
Simplify E fully.
Evaluate E at the stated inputs \(\displaystyle x=4, y=3\).
Find the constant term.
Evaluate E at \(\displaystyle x=4, y=3\).
Show the reasoning
(a)
Collect the x terms, then the y terms, retaining the constant: \(6 x + 4 y + 28\).
Answer: \(6 x + 4 y + 28\)
(b)
Substitute into the collected expression: \((6)(4)+(4)(3)+28\) = \(64\).
Answer: \(64\)
(c)
Collect coefficients of identical variable terms, or substitute the stated inputs into the whole simplified expression.
\(28\)
Answer: \(28\).
(d)
Collect coefficients of identical variable terms, or substitute the stated inputs into the whole simplified expression.
\(64\)
Answer: \(64\).
Example 2 · Expressions, substitution and like terms
Consider \(E=8 x + 7 y + 29\).
Simplify E fully.
Evaluate E at the stated inputs \(\displaystyle x=5, y=1\).
Find the constant term.
Evaluate E at \(\displaystyle x=5, y=1\).
Show the reasoning
(a)
Collect the x terms, then the y terms, retaining the constant: \(8 x + 7 y + 29\).
Answer: \(8 x + 7 y + 29\)
(b)
Substitute into the collected expression: \((8)(5)+(7)(1)+29\) = \(76\).
Answer: \(76\)
(c)
Collect coefficients of identical variable terms, or substitute the stated inputs into the whole simplified expression.
\(29\)
Answer: \(29\).
(d)
Collect coefficients of identical variable terms, or substitute the stated inputs into the whole simplified expression.
\(76\)
Answer: \(76\).
Example 3 · Expressions, substitution and like terms
Consider \(E=6 x + 7 y + 30\).
Simplify E fully.
Evaluate E at the stated inputs \(\displaystyle x=2, y=2\).
Find the constant term.
Evaluate E at \(\displaystyle x=2, y=2\).
Show the reasoning
(a)
Collect the x terms, then the y terms, retaining the constant: \(6 x + 7 y + 30\).
Answer: \(6 x + 7 y + 30\)
(b)
Substitute into the collected expression: \((6)(2)+(7)(2)+30\) = \(56\).
Answer: \(56\)
(c)
Collect coefficients of identical variable terms, or substitute the stated inputs into the whole simplified expression.
\(30\)
Answer: \(30\).
(d)
Collect coefficients of identical variable terms, or substitute the stated inputs into the whole simplified expression.
\(56\)
Answer: \(56\).
Common mistakes
Keep intermediate values exact until the final step.