C2.2; E2.2 - Collecting like terms

C2.2; E2.2 - Collecting like terms

Two pieces of ribbon have lengths 4x4x cm and 3x3x cm. Their total is 7x7x cm, but what changes if the second length is 3y3y cm? In C2.2 and E2.2, you will simplify expressions by combining matching terms while keeping different variable parts separate.

Like terms

The opening sum 4x+3x4x+3x contains seven copies of the same number xx, so it equals 7x7x. This works for every value of xx. Adding the coefficients counts how many copies there are; it does not change what is being counted.

Term, coefficient and like terms

A term is a number or product separated from other terms by addition or subtraction. Its coefficient is its numerical multiplier, including its sign. Like terms have exactly the same variable part, including the powers of every letter.

In 6x−2x+x6x-2x+x, the coefficients are 6, −2-2 and 1. The unwritten coefficient of xx is 1. All three terms are like terms, so their coefficients can be combined.

Worked example 1: adding matching terms

Question. Simplify 5x+2x5x+2x.

Plan. Add the coefficients because both terms are multiples of xx.

5x+2x=(5+2)xfive copies plus two copies=7x\begin{aligned} 5x+2x&=(5+2)x &&\text{five copies plus two copies}\\ &=7x \end{aligned}

Answer. 7x7x.

Check. At x=3x=3, the original is 15+6=2115+6=21, and 7x=7×3=217x=7\times3=21.

The next expression changes the joining sign. Subtracting two copies reduces the coefficient.

Worked example 2: subtracting matching terms

Question. Simplify 5x−2x5x-2x.

Plan. Combine the signed coefficients, leaving the variable part unchanged.

5x−2x=(5−2)xsubtract two copies=3x\begin{aligned} 5x-2x&=(5-2)x &&\text{subtract two copies}\\ &=3x \end{aligned}

Answer. 3x3x.

Check. At x=3x=3, the original is 15−6=915-6=9, and the answer gives 3×3=93\times3=9.

Collecting means changing the form of an expression, not finding a value of its letter. There is no equation to solve.

Watch out

5x+2x=7x5x+2x=7x, not 7x27x^2. Addition counts copies; multiplication such as x×xx\times x creates a square. Also, x−x=0x-x=0, not xx.

Remember the unwritten coefficient in the next expression.

Several variables

Different letters may represent different numbers. You cannot combine 4x+3y4x+3y into one multiple of xx or yy unless an extra relationship is given. The expression is already simplified.

With several letters, first group terms that match. Move each term together with its sign: −3b-3b stays negative wherever it is placed.

Worked example 3: grouping two kinds of term

Question. Simplify 4a+3b+2a+b4a+3b+2a+b.

Plan. Collect the aa-terms and bb-terms separately.

4a+3b+2a+b=4a+2a+3b+bgroup like terms=(4+2)a+(3+1)bcombine each coefficient group=6a+4b\begin{aligned} 4a+3b+2a+b&=4a+2a+3b+b &&\text{group like terms}\\ &=(4+2)a+(3+1)b &&\text{combine each coefficient group}\\ &=6a+4b \end{aligned}

Answer. 6a+4b6a+4b.

Check. At a=2a=2 and b=3b=3, the original gives 8+9+4+3=248+9+4+3=24. The simplified expression gives 12+12=2412+12=24.

Now change only the sign of 3b3b. Carry that sign when regrouping.

Worked example 4: a negative coefficient

Question. Simplify 4a−3b+2a+b4a-3b+2a+b.

Plan. Add the aa-coefficients, then add the signed bb-coefficients.

4a−3b+2a+b=4a+2a−3b+b=(4+2)a+(−3+1)binclude the negative sign=6a−2b\begin{aligned} 4a-3b+2a+b&=4a+2a-3b+b\\ &=(4+2)a+(-3+1)b &&\text{include the negative sign}\\ &=6a-2b \end{aligned}

Answer. 6a−2b6a-2b.

Check. At a=2a=2, b=3b=3, the original is 8−9+4+3=68-9+4+3=6; the simplified form is 12−6=612-6=6.

Substitution can expose a wrong simplification, although one matching numerical result does not prove that two expressions are always equal. Grouping matching terms is the general justification.

Watch out

4a−3b+2a+b4a-3b+2a+b does not become 6a−4b6a-4b. The last term is +b+b, so the bb-coefficient is −3+1=−2-3+1=-2. A subtraction sign belongs to the term immediately following it unless brackets say otherwise.

Keep the two variable groups separate.

Powers and constants

A variable part includes its powers, not just its letters. The terms xx and x2x^2 are different: at x=3x=3, they are 3 and 9. The terms xyxy and x2x^2 also differ, while xyxy and yxyx match because multiplication can be reordered.

Constant term

A constant term has no variable part. Constants can be collected with other constants. For example, the constants in x2+5−x−7x^2+5-x-7 are +5+5 and −7-7, giving −2-2.

To avoid missing a group, collect squared terms, then other matching variable terms, then constants. Writing powers in descending order makes the result easy to read.

Worked example 5: squares, linear terms and constants

Question. Simplify 3x2+2x+5+x2−7+x3x^2+2x+5+x^2-7+x.

Plan. Keep x2x^2, xx and the constant terms in separate groups.

3x2+2x+5+x2−7+x=(3+1)x2+(2+1)x+(5−7)=4x2+3x−2combine each group\begin{aligned} 3x^2+2x+5+x^2-7+x&=(3+1)x^2+(2+1)x+(5-7)\\ &=4x^2+3x-2 &&\text{combine each group} \end{aligned}

Answer. 4x2+3x−24x^2+3x-2.

Check. At x=2x=2, the original gives 12+4+5+4−7+2=2012+4+5+4-7+2=20. The answer gives 16+6−2=2016+6-2=20.

The next example replaces the xx-terms with xyxy-terms. Their coefficients still combine, while the squared terms remain a different group.

Worked example 6: mixed-variable terms

Question. Simplify 3x2+2xy+5+x2−7+xy3x^2+2xy+5+x^2-7+xy.

Plan. Collect the x2x^2-terms, xyxy-terms and constants separately.

3x2+2xy+5+x2−7+xy=(3+1)x2+(2+1)xy+(5−7)=4x2+3xy−2\begin{aligned} 3x^2+2xy+5+x^2-7+xy&=(3+1)x^2+(2+1)xy+(5-7)\\ &=4x^2+3xy-2 \end{aligned}

Answer. 4x2+3xy−24x^2+3xy-2.

Check. At x=2x=2, y=3y=3, the original is 12+12+5+4−7+6=3212+12+5+4-7+6=32. The answer gives 16+18−2=3216+18-2=32.

This is the same principle as the ribbons: only matching quantities can be counted together. The surprising part is how little the rule changes when more letters and powers appear.

Watch out

x2+xx^2+x is not 2x22x^2, and x2+xyx^2+xy is not 2x2y2x^2y. Their variable parts differ. Simplest form may contain several terms; do not force them into one.

Finish by checking all three groups.

Quick Recap

  • Like terms have the same letters with the same powers.
  • Combine signed coefficients and keep the variable part unchanged.
  • Treat constants as a separate group; remember the unwritten coefficient 1.
  • Several unlike terms may remain in simplest form.

The ribbon sum is 7x7x cm when both pieces are multiples of xx. With lengths 4x4x cm and 3y3y cm, the total stays 4x+3y4x+3y cm unless a relationship between xx and yy is known.