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Substitution

Algebra · level 4, lesson 2 of 7

A question to start

Most of the world measures temperature in degrees Celsius (°C). In the United States people use degrees Fahrenheit (°F). The rule for changing Celsius into Fahrenheit is

\[F = \frac{9C}{5} + 32.\]

Water boils at 100 °C, which is \(\dfrac{9 \times 100}{5} + 32 = 180 + 32 = 212\) °F. Is there a temperature that is the same number in both scales?

There is: \(-40\). Put \(C = -40\) into the rule: \(\dfrac{9 \times (-40)}{5} + 32 = \dfrac{-360}{5} + 32 = -72 + 32 = -40\). So \(-40\) °C and \(-40\) °F are exactly as cold as each other. To find surprises like this, you need one skill: putting numbers into formulas correctly, even negative ones. That skill is called substitution.

The idea

Substitute means "put in place of"

To substitute a number into an expression means to put the number in place of the letter, and then work out the answer. The answer is called the value of the expression. Working it out is called evaluating.

For \(3n + 2\) with \(n = 5\):

\[3n + 2 = 3 \times 5 + 2 = 15 + 2 = 17.\]

Notice that when the letter goes away, the hidden times sign comes back. \(3n\) means \(3 \times n\), so it becomes \(3 \times 5\), not "35".

Follow the order of operations

After substituting, you have an ordinary calculation, and the usual order of operations applies:

  1. Brackets first.
  2. Then powers (squares and so on).
  3. Then multiply and divide, from left to right.
  4. Then add and subtract, from left to right.

So for \(2(x + 3)^2\) with \(x = 1\): brackets first, \(1 + 3 = 4\); then the square, \(4^2 = 16\); then multiply, \(2 \times 16 = 32\).

Put a negative number in brackets

Here is the most important habit in this lesson. When you substitute a negative number, write it inside brackets. Look at \(n^2\) with \(n = -3\):

  • Right: \((-3)^2 = (-3) \times (-3) = 9\).
  • Wrong: \(-3^2\). By the order of operations the square comes before the minus sign, so \(-3^2\) means \(-(3^2) = -9\).

The brackets keep the minus sign attached to the 3, so the whole number \(-3\) gets squared. Why is \((-3) \times (-3)\) positive? Look at this pattern:
\(3 \times (-3) = -9\) \(2 \times (-3) = -6\) \(1 \times (-3) = -3\)
\(0 \times (-3) = 0\) \((-1) \times (-3) = 3\) \((-2) \times (-3) = 6\)

Each time the first number goes down by 1, the answer goes up by 3. Following the pattern past zero gives \((-3) \times (-3) = 9\). A negative times a negative is positive.

Quick rules for multiplying and dividing:

  • positive × positive = positive; negative × negative = positive;
  • positive × negative = negative; negative × positive = negative.

The same rules work for division.

Substituting into formulas

A formula is an equation that tells you how to work out one quantity from others, like \(F = \dfrac{9C}{5} + 32\) or \(P = 2l + 2w\). Each letter has a meaning and a unit. To use a formula:

  1. Write the formula.
  2. Write it again with the numbers in place of the letters (negatives in brackets).
  3. Work it out, step by step.
  4. Write the answer with its unit, and ask if it makes sense.

Substituting more than one letter

Expressions can have several letters. For \(3a - 2b\) with \(a = 4\) and \(b = -5\):

\[3 \times 4 - 2 \times (-5) = 12 - (-10) = 12 + 10 = 22.\]

Taking away a negative is the same as adding: if you remove a debt of 10 dollars, you are 10 dollars better off.

Why substitution matters

Substitution is how you use algebra: you write a rule once, then put in any numbers you like. It is also how you check algebra: after solving an equation, you substitute the answer back in. And it is how you test a claim: if someone says \((a + b)^2 = a^2 + b^2\), substitute \(a = 1\), \(b = 1\): the left side is \(2^2 = 4\), the right side is \(1 + 1 = 2\). One example is enough to show the claim is false.

Worked examples

1. A simple expression. Find \(5x - 7\) when \(x = 3\), and when \(x = -2\).

  • \(x = 3\): \(5 \times 3 - 7 = 15 - 7 = 8\).
  • \(x = -2\): \(5 \times (-2) - 7 = -10 - 7 = -17\).

2. Brackets and powers. Find \((y - 4)^2\) and \(y^2 - 4\) when \(y = 1\).

  • \((1 - 4)^2 = (-3)^2 = 9\).
  • \(1^2 - 4 = 1 - 4 = -3\).
  • Different answers: the brackets change what gets squared.

3. A formula with two letters. The perimeter of a rectangle is \(P = 2l + 2w\). Find \(P\) when \(l = 8.5\) cm and \(w = 3\) cm.

\(P = 2 \times 8.5 + 2 \times 3 = 17 + 6 = 23\) cm.

4. Temperature on Mars. Nights on Mars are very cold; temperatures around \(-80\) °C are common. What is \(-80\) °C in Fahrenheit?

\(F = \dfrac{9 \times (-80)}{5} + 32 = \dfrac{-720}{5} + 32 = -144 + 32 = -112\) °F.

5. Distance. A car travels at a steady speed. Distance = speed × time, or \(d = st\). Find \(d\) when \(s = 60\) km per hour and \(t = 2.5\) hours.

\(d = 60 \times 2.5 = 150\) km.

6. Negative times negative. Find \(a^2 - 3ab\) when \(a = -2\) and \(b = 4\).

\((-2)^2 - 3 \times (-2) \times 4 = 4 - (-24) = 4 + 24 = 28\).

Try it

  1. Find \(4m + 9\) when \(m = 6\).
  2. Find \(10 - 3k\) when \(k = 5\), and when \(k = -5\).
  3. Find \(n^2 + 1\) when \(n = -4\).
  4. Find \(2(x + 7)\) and \(2x + 7\) when \(x = 3\).
  5. Use \(F = \dfrac{9C}{5} + 32\) to change 25 °C (a warm day) into Fahrenheit, and 0 °C.
  6. Find \(\dfrac{p + q}{2}\) when \(p = 13\) and \(q = -5\).
  7. The area of a triangle is \(A = \dfrac{bh}{2}\), with \(b\) the base and \(h\) the height. Find \(A\) when \(b = 9\) m and \(h = 4\) m.
  8. Priya says \((a + b)^2\) is always equal to \(a^2 + b^2\). Substitute \(a = 2\) and \(b = 3\) to test her claim.
Answers
  1. \(4 \times 6 + 9 = 24 + 9 = 33\).
  2. \(k = 5\): \(10 - 15 = -5\). \(k = -5\): \(10 - 3 \times (-5) = 10 + 15 = 25\).
  3. \((-4)^2 + 1 = 16 + 1 = 17\).
  4. \(2(3 + 7) = 2 \times 10 = 20\), and \(2 \times 3 + 7 = 13\).
  5. 25 °C: \(\dfrac{9 \times 25}{5} + 32 = 45 + 32 = 77\) °F. 0 °C: \(0 + 32 = 32\) °F (water freezes).
  6. \(\dfrac{13 + (-5)}{2} = \dfrac{8}{2} = 4\). (This is the number halfway between \(-5\) and 13.)
  7. \(A = \dfrac{9 \times 4}{2} = \dfrac{36}{2} = 18\) square metres.
  8. \((2 + 3)^2 = 5^2 = 25\), but \(2^2 + 3^2 = 4 + 9 = 13\). They are not equal, so the claim is false. (It is true only in special cases, like when \(a\) or \(b\) is 0.)

Common mistake

Squaring a negative without brackets. With \(n = -5\), \(n^2\) is \((-5)^2 = 25\). Writing \(-5^2\) on a calculator gives \(-25\), because the calculator squares the 5 first. Always type and write the brackets.

Gluing the number to the letter's coefficient. With \(n = 4\), \(3n\) is \(3 \times 4 = 12\), not 34.

Where it leads