Substituting means putting a number where the letter stands
A number beside a letter means multiply
In algebra a letter stands for a number. A number written just in front of a letter means multiply, so 3n means 3 × n, which is 3 lots of whatever n is. Draw it as a rectangle 3 wide and n tall: its area is 3 × n, or 3n.
A rectangle 3 wide and n tall has area 3 × n, written 3n.
Put the number in place of the letter
Once we know what n is, we can find the value of 3n. If n = 4, replace n with 4: 3n becomes 3 × 4 = 12. Replacing a letter with its value is called substitution.
When the letter goes, put the multiplication sign back. 3n with n = 4 is 3 × 4, which is 12. It is not 34, and it is not 3 + 4.
With n = 4 the rectangle is 3 by 4, so its area is 12.
Only the letter changes
Now find the value of 3n + 2 when n = 4. Think of the letter as a box. The 4 goes into the box where n stands, and only there. The 3 in front and the + 2 after it stay exactly as they were: 3 × 4 + 2.
Then work it out with the order of operations, multiplication before addition: 3 × 4 = 12, and 12 + 2 = 14. So when n = 4, 3n + 2 = 14.
The 4 goes into the box for n. The 3 and the + 2 do not change.
The value of n is multiplied by 3 first, and then 2 is added, which gives 14.
Multiply before you add
The order matters. Adding first gives (4 + 2) × 3 = 18, which is the value of a different expression, 3(n + 2). In 3n + 2, only n is multiplied by 3.
A power comes before the multiplication in front of it. means , so n is squared first. When n = 5, . It is not .
The same number in every place
A letter can appear more than once in an expression. Wherever it appears, it stands for the same number, so the value goes into every box. When n = 5, .
Both boxes for n take the same value, 5.
Negative values go in brackets
When the value is negative, write it in brackets as it goes in. That keeps the minus sign with the number. When n = −2, 3n + 3 = 3 × (−2) + 3. A positive number times a negative number is negative, so 3 × (−2) = −6, and −6 + 3 = −3.
Brackets matter most with a power. When n = −3, , because a negative number times a negative number is positive.
The value −2 goes into the box with its minus sign, inside brackets.
The usual mistakes
Reading 3n as 3 + n. A number next to a letter means multiply, so with n = 4, 3n is 12, not 7.
Adding before multiplying. In 3n + 2 the 2 is added after n has been multiplied by 3, so with n = 4 the value is 14, not 18.
Algebraic expressions · A Temperature Rule Written as an Expression and Evaluated
QuestionA weather station changes a temperature of c °C into degrees Fahrenheit by this rule: multiply the temperature by 95, then add 32. (a) Write the rule as an expression in c, and find the Fahrenheit temperature when c = 25. (b) On a winter night c = −10. Find the Fahrenheit temperature.
1.Multiplying c by 95 gives 95c, and adding 32 gives the expression 95c + 32.
Multiply c by 95, then add 32: the expression is 95c + 32. 2.Substitute c = 25: 95 × 25 + 32. Multiply before adding: 95 × 25 = 9 × 5 = 45.
Substitute c = 25 and multiply first: 95 × 25 = 45. 3.(a) The expression is 95c + 32. When c = 25 its value is 45 + 32 = 77, so the temperature is 77 °F.
(a) 45 + 32 = 77, so 25 °C is 77 °F. 4.Substitute c = −10 with brackets round the negative number: 95 × (−10) + 32. A positive number times a negative number is negative, so 95 × (−10) = −18.
Substitute c = −10 in brackets: 95 × (−10) = −18. 5.(b) −18 + 32 = 14, so the temperature is 14 °F. Check: from 25 °C down to −10 °C is a fall of 35 °C, which is 95 × 35 = 63 °F, and 77 − 63 = 14.
(b) −18 + 32 = 14, so −10 °C is 14 °F.
Answer: (a) 95c + 32, which is 77 °F; (b) 14 °F
Watch out for
- Adding before multiplying, as in 95 × (25 + 32). The rule multiplies the temperature first and adds 32 afterwards, so only c is multiplied by 95.
- Writing 95 × (−10) as +18. A positive number times a negative number is negative, so the product is −18 and the answer is −18 + 32 = 14, not 50.
Your turn
1.If n = −2, what is 3n + 3?