2.2, 2.3 - Expressions, substitution and like terms

2.2, 2.3 - Expressions, substitution and like terms

In 2.3 you will read conventional algebraic notation and replace letters by given values. In 2.2 you will evaluate those expressions and collect like terms. Careful brackets and signs let you change the form without changing its meaning.

Roles and algebraic notation

Can a square be negative just because the number being squared is negative? A calculator entry such as −22-2^2 can give −4-4. We will see why this does not make the square of −2-2 negative.

You already know

A letter represents a number. Multiplication and division come before addition and subtraction; powers come before multiplication. Brackets group what must stay together.

A variable can take different values. An unknown is a number to find from a stated relationship. Thus tt may vary in 3t+53t+5, but is fixed by 3t+5=203t+5=20. A formula such as C=rn+fC=rn+f relates quantities: a rate rr, a count nn, a fixed fee ff, and a total CC. A formula is an equation; an expression has no equals sign.

Conventional algebraic notation

Write 3x3x for 3×x3\times x, abab for a×ba\times b, x2x^2 for x×xx\times x, and xy\frac{x}{y} for division by yy, where y≠0y\ne0. A fraction bar groups the whole numerator and denominator.

For example, “five more than triple tt” is 3t+53t+5, while “triple the sum of tt and 5” is 3(t+5)3(t+5). At t=2t=2, they give 11 and 21. The brackets change which quantity is tripled. Similarly, abab is a product, not a+ba+b. These notation rules are not supplied in either tier's formulae sheet: learn them.

Key idea

Read the operations and grouping before doing any calculation. The letter's role comes from its situation, not its name.

A denominator must be nonzero because division by zero is undefined. This restriction still applies when you write the division as a fraction bar.

Substituting negative values

Substitution turns an expression into an ordinary numerical calculation. Replace every occurrence of the letter, then follow the usual operation order.

Substitution

Substitution means replacing a letter by its given value. If x=−2x=-2, replace x2x^2 by (−2)2(-2)^2. The brackets show that the negative sign belongs to the value being squared.

The coefficient still multiplies: 3x=3(−2)=−63x=3(-2)=-6. In −x2-x^2, the minus is outside the square, so −x2=−(−2)2=−4-x^2=-(-2)^2=-4. That is different from squaring a negative number. The substitution and sign rules are not supplied in either tier's formulae sheet: learn them.

Worked example 1: one negative input

Question. Evaluate 2x2−3y2x^2-3y when x=−2x=-2 and y=3y=3.

Plan. Substitute in brackets, evaluate the square, then multiply and subtract.

2x2−3y=2(−2)2−3(3)replace both letters=2(4)−9square before multiplying=8−9=−1subtract\begin{aligned} 2x^2-3y&=2(-2)^2-3(3) &&\text{replace both letters}\\ &=2(4)-9 &&\text{square before multiplying}\\ &=8-9=-1 &&\text{subtract} \end{aligned}

Why this step? The whole value −2-2 replaces xx, so its sign is inside the square.

Answer. −1-1.

Check. The first term is 8 and the amount subtracted is 9, so the answer should be just below zero.

Change only the sign of yy. This makes the second term positive after the subtraction is carried out.

Worked example 2: subtracting a negative term

Question. Evaluate 2x2−3y2x^2-3y when x=−2x=-2 and y=−3y=-3.

Plan. Substitute both negative values in brackets and calculate the two terms.

2x2−3y=2(−2)2−3(−3)substitute=8−(−9)evaluate each product=8+9=17subtract a negative\begin{aligned} 2x^2-3y&=2(-2)^2-3(-3) &&\text{substitute}\\ &=8-(-9) &&\text{evaluate each product}\\ &=8+9=17 &&\text{subtract a negative} \end{aligned}

Why this step? We are subtracting −9-9, so the value increases by 9.

Answer. 17.

Check. Changing yy from 3 to −3-3 increases −3y-3y by 18. The answers differ by 17−(−1)=1817-(-1)=18.

A repeated letter always uses the same supplied value. It helps to finish the entire substitution line before calculating anything.

Watch out

The line −a2=(−a)2-a^2=(-a)^2 is wrong. The minus on the left is outside the power. At a=−3a=-3, the correct values are −(−3)2=−9-(-3)^2=-9 and (−(−3))2=9(-(-3))^2=9.

Try the same expression with a new value of xx. Keep the brackets visible in your first line, including in a calculator entry.

Fractions, decimals and formulae

Fractional and decimal inputs replace letters in exactly the same way. A useful surprise is that squaring can make a positive number smaller: (12)2=14<12(\frac12)^2=\frac14<\frac12. Multiplying by a positive number below 1 reduces a positive value.

For a formula involving measurements, first check that the units agree. Convert them before substitution if needed. A denominator that becomes zero makes the expression undefined. Keep fractions exact unless a rounded answer is requested.

Rectangle perimeter

P=2(l+w)P=2(l+w)

Here ll and ww are the rectangle's length and width in the same units, and PP is its perimeter in those units. The four sides give l+w+l+wl+w+l+w.

Not in either tier's formulae sheet: learn it.

Substitution also works in a formula stated in words. “Perimeter is twice the sum of length and width” gives exactly the calculation below.

Worked example 3: a fractional width

Question. A rectangle has l=3.5 cml=3.5\,\mathrm{cm} and w=34 cmw=\frac34\,\mathrm{cm}. Use P=2(l+w)P=2(l+w) to find its perimeter.

Plan. Use compatible centimetre values and convert the terminating fraction exactly.

P=2(3.5+34)substitute lengths=2(3.5+0.75)use an exact decimal=2(4.25)=8.5 cmadd then double\begin{aligned} P&=2\left(3.5+\frac34\right) &&\text{substitute lengths}\\ &=2(3.5+0.75) &&\text{use an exact decimal}\\ &=2(4.25)=8.5\,\mathrm{cm} &&\text{add then double} \end{aligned}

Why this step? The bracket is the sum of two adjacent sides; doubling counts all four.

Answer. 8.5 cm8.5\,\mathrm{cm}, or 172 cm\frac{17}{2}\,\mathrm{cm}.

Check. 3.5+0.75+3.5+0.75=8.53.5+0.75+3.5+0.75=8.5.

Now change only the width's unit. The physical width is still the same, but it must be converted before it joins the length.

Worked example 4: converting before substitution

Question. A rectangle has l=3.5 cml=3.5\,\mathrm{cm} and w=7.5 mmw=7.5\,\mathrm{mm}. Use P=2(l+w)P=2(l+w) to find its perimeter in centimetres.

Plan. Convert millimetres to centimetres, then substitute.

w=7.5÷10=0.75 cmconvert the widthP=2(3.5+0.75)substitute matching units=8.5 cmevaluate\begin{aligned} w&=7.5\div10=0.75\,\mathrm{cm} &&\text{convert the width}\\ P&=2(3.5+0.75) &&\text{substitute matching units}\\ &=8.5\,\mathrm{cm} &&\text{evaluate} \end{aligned}

Why this step? Adding 3.53.5 and 7.57.5 directly would combine counts of different-sized units.

Answer. 8.5 cm8.5\,\mathrm{cm}.

Check. In millimetres, 2(35+7.5)=85 mm=8.5 cm2(35+7.5)=85\,\mathrm{mm}=8.5\,\mathrm{cm}.

With powers, replace the whole fraction: (12)2=14(\frac12)^2=\frac14, not 12\frac12. For another example, at p=−2p=-2, q=0.6q=0.6, the expression 32p2−q\frac32p^2-q gives 32(4)−0.6=5.4\frac32(4)-0.6=5.4.

Watch out

The line 2(3.5+7.5)=22 cm2(3.5+7.5)=22\,\mathrm{cm} is wrong for the second rectangle. The width 7.5 was in millimetres. Convert it to 0.75 cm0.75\,\mathrm{cm} first, giving 8.5 cm8.5\,\mathrm{cm}.

The next rectangle changes only the width's value. State the requested unit with your final answer.

Collecting like terms

Substitution finds a value; simplifying writes a shorter expression that works for every allowed input. Three groups of xx plus two groups of xx make five groups of xx. The value of xx does not need to be known.

Like terms

Like terms have the same variable factors with the same powers. Thus 2ab2ab and 3ba3ba are like terms because ab=baab=ba. The terms 2x2x and 3x23x^2 are not like terms; neither are 2x2x and 3y3y. Constants collect with constants.

Add or subtract the coefficients and keep the variable part. This follows from the distributive law: 2x+5x=(2+5)x=7x2x+5x=(2+5)x=7x. A coefficient may be negative or unwritten: −x2-x^2 has coefficient −1-1. Collection rules are not supplied in either tier's formulae sheet: learn them.

Worked example 5: collecting three groups

Question. Simplify 3x2−4x+5+2x2+7x−83x^2-4x+5+2x^2+7x-8.

Plan. Group square terms, linear terms and constants, keeping their signs.

3x2−4x+5+2x2+7x−8=(3+2)x2+(−4+7)x+(5−8)group like terms=5x2+3x−3add signed coefficients\begin{aligned} &3x^2-4x+5+2x^2+7x-8\\ &=(3+2)x^2+(-4+7)x+(5-8) &&\text{group like terms}\\ &=5x^2+3x-3 &&\text{add signed coefficients} \end{aligned}

Why this step? Each group shares one variable part; different groups cannot be combined.

Answer. 5x2+3x−35x^2+3x-3.

Check. At x=−2x=-2, the original gives 12+8+5+8−14−8=1112+8+5+8-14-8=11. The answer gives 20−6−3=1120-6-3=11.

Change only the sign of the second linear term. Attach that sign to the term before moving it.

Worked example 6: keeping a changed sign

Question. Simplify 3x2−4x+5+2x2−7x−83x^2-4x+5+2x^2-7x-8.

Plan. Collect the same three groups, including the negative coefficient of −7x-7x.

3x2−4x+5+2x2−7x−8=(3+2)x2+(−4−7)x+(5−8)keep each sign=5x2−11x−3combine coefficients\begin{aligned} &3x^2-4x+5+2x^2-7x-8\\ &=(3+2)x^2+(-4-7)x+(5-8) &&\text{keep each sign}\\ &=5x^2-11x-3 &&\text{combine coefficients} \end{aligned}

Why this step? The sign belongs to the whole term −7x-7x, wherever that term is written.

Answer. 5x2−11x−35x^2-11x-3.

Check. At x=1x=1, both forms give −9-9.

If coefficients add to zero, that term disappears. Also distinguish sums from products: x2+x2=2x2x^2+x^2=2x^2, while x2×x2=x4x^2\times x^2=x^4.

Watch out

The line 3x2+2x=5x23x^2+2x=5x^2 is wrong because x2x^2 and xx are different variable parts. The correct simplified expression is still 3x2+2x3x^2+2x. At x=2x=2, it is 16, whereas 5x2=205x^2=20.

We can now answer the opening question. The square of −2-2 is (−2)2=4(-2)^2=4. The different expression −22-2^2 squares 2 first, then negates the result, giving −4-4.

Quick Recap

  • Read products, powers and fraction bars using conventional notation.
  • Replace every letter by its full value, using brackets around negative and fractional inputs.
  • Use compatible units and avoid rounding exact fractions too early.
  • Collect only identical variable parts, keeping each coefficient's sign.

A negative number's square is nonnegative: (−2)2=4(-2)^2=4. The calculator entry −22-2^2 means the different calculation −(22)=−4-(2^2)=-4.