Pre-Algebra · Class 12
Evaluating Expressions and Combining Like Terms
To evaluate an expression, substitute the given values and follow the order of operations. To combine like terms, add or subtract their coefficients while keeping the variables and exponents unchanged. The distributive property connects these skills when parentheses are involved.
This SOMATH Pre-Algebra Class 12 lesson contains ten images in sequence, five progressively harder questions after each image, and ten final word problems. All 60 questions are new practice rather than copies of the poster examples, with worked solutions behind compact blue Answer buttons.
For guided study, view the SOMATH schedule or book an evaluation. Work through one poster and its five questions before moving to the next topic.
Parts of an algebraic expression
An expression represents a value. In an expanded sum, a term includes its sign; a coefficient multiplies a variable part, and a constant has no variable. An unwritten coefficient is still present: a lone variable has coefficient one, and its negative has coefficient negative one.
Try it yourself: five questions, increasing difficulty
Q1 Practice 1 of 5 · Foundation
List the terms in \(7m+2\). Which term is constant?
Answer
Terms: \(7m\) and \(2\). Constant: \(2\).
- The plus sign separates the two terms. Only \(7m\) contains a variable.
Q2 Practice 2 of 5 · Build
In \(-4a+9\), identify the variable, its coefficient, and the constant.
Answer
Variable: \(a\). Coefficient: \(-4\). Constant: \(9\).
- The coefficient includes the minus sign because \(-4\) multiplies \(a\).
Q3 Practice 3 of 5 · Apply
List the three terms and the two variable coefficients in \(8p-3q+6\).
Answer
Terms: \(8p,-3q,6\). Coefficients: \(8,-3\).
- Keep the sign attached to each term. The number six is the constant, not a variable coefficient.
Q4 Practice 4 of 5 · Reason
Identify all variable coefficients in \(-x+5y-11\). How many terms are there?
Answer
Coefficients: \(-1,5\). There are three terms.
- Rewrite \(-x\) as \(-1x\). The terms are \(-x,5y,-11\).
Q5 Practice 5 of 5 · Challenge
For \(4a^2-2ab+b-9\), list all terms, their variable coefficients, and the distinct variables.
Answer
Terms: \(4a^2,-2ab,b,-9\). Variable coefficients: \(4,-2,1\). Variables: \(a,b\).
- The term \(-2ab\) is one term containing a product of two variables.
- The exponent two belongs to the variable part of \(4a^2\), not to its coefficient. The constant is \(-9\).
Evaluating expressions by substitution
To evaluate, replace every occurrence of each variable with its given value, then calculate. A coefficient next to a variable means multiplication, not a two-digit number. In a fraction, the entire numerator is grouped together.
Try it yourself: five questions, increasing difficulty
Q6 Practice 1 of 5 · Foundation
Evaluate \(6x-5\) when \(x=4\).
Answer
\(19\).
- \(6(4)-5=24-5=19\).
Q7 Practice 2 of 5 · Build
Evaluate \(2a+3b\) when \(a=5\) and \(b=2\).
Answer
\(16\).
- Substitute both values: \(2(5)+3(2)=10+6=16\).
Q8 Practice 3 of 5 · Apply
Evaluate \(\frac m4+2\) when \(m=12\).
Answer
\(5\).
- \(\frac{12}{4}+2=3+2=5\). Divide before adding.
Q9 Practice 4 of 5 · Reason
Evaluate \(3u^2-v\) when \(u=2\) and \(v=5\).
Answer
\(7\).
- Square first: \(3(2^2)-5=3(4)-5=12-5=7\).
Q10 Practice 5 of 5 · Challenge
Evaluate \(\frac{2p+q}{3}\) when \(p=4\) and \(q=7\).
Answer
\(5\).
- The numerator is \(2(4)+7=15\).
- Divide the whole numerator by three: \(15/3=5\), not \(8+7/3\).
Following the order of operations
Work inside grouping symbols, then evaluate exponents. Multiplication and division share priority and are performed left to right; addition and subtraction also share priority and proceed left to right. Parentheses can change the value of an expression, so do not remove them casually.
Try it yourself: five questions, increasing difficulty
Q11 Practice 1 of 5 · Foundation
Evaluate \(4+3x\) when \(x=5\).
Answer
\(19\).
- \(4+3(5)=4+15=19\). Multiply before adding.
Q12 Practice 2 of 5 · Build
Evaluate \(5(x-2)\) when \(x=6\).
Answer
\(20\).
- First calculate inside parentheses: \(5(6-2)=5(4)=20\).
Q13 Practice 3 of 5 · Apply
Evaluate \(3x^2-2\) when \(x=4\).
Answer
\(46\).
- The square comes before multiplication: \(3(16)-2=48-2=46\).
Q14 Practice 4 of 5 · Reason
Evaluate \(24\div x\times2+1\) when \(x=3\).
Answer
\(17\).
- Division and multiplication have equal priority. Work left to right: \(24\div3=8\), then \(8\times2=16\).
- Finally add one. Do not treat this as \(24\div(3\times2)+1\).
Q15 Practice 5 of 5 · Challenge
Evaluate \(2[3+(x-1)^2]-12\div3\) when \(x=4\).
Answer
\(20\).
- Inside the brackets, \(x-1=3\), so \(3+(x-1)^2=3+9=12\).
- Then \(2(12)-12\div3=24-4=20\).
Substituting negative numbers
Put a negative value inside parentheses whenever you substitute it. Squaring the entire negative number produces a positive result, while a minus sign outside a square is applied after the square. Subtracting a negative is the same as adding its opposite.
Try it yourself: five questions, increasing difficulty
Q16 Practice 1 of 5 · Foundation
Evaluate \(x+9\) when \(x=-5\).
Answer
\(4\).
- \((-5)+9=4\).
Q17 Practice 2 of 5 · Build
Evaluate \(-3x+2\) when \(x=-4\).
Answer
\(14\).
- \(-3(-4)+2=12+2=14\). Two negative factors give a positive product.
Q18 Practice 3 of 5 · Apply
Evaluate \(x^2+3x\) when \(x=-6\).
Answer
\(18\).
- \((-6)^2+3(-6)=36-18=18\).
Q19 Practice 4 of 5 · Reason
Evaluate \(-x^2+2x\) when \(x=-3\). Explain the first minus sign.
Answer
\(-15\).
- The first term is \(-[(-3)^2]=-9\), not \(9\).
- The second term is \(2(-3)=-6\), so the total is \(-9-6=-15\).
Q20 Practice 5 of 5 · Challenge
Evaluate \(2a^2-3ab+b\) when \(a=-2\) and \(b=3\).
Answer
\(29\).
- \(2(-2)^2=8\). Also, \(-3(-2)(3)=18\).
- Add all terms: \(8+18+3=29\).
Identifying like and unlike terms
Like terms have exactly the same variable part, including every exponent. Their coefficients can differ. All constants are like terms. Reordering factors does not change their product, but changing which variable is squared changes the term.
Try it yourself: five questions, increasing difficulty
Q21 Practice 1 of 5 · Foundation
Are \(3m\) and \(8m\) like terms? Explain.
Answer
Yes.
- Both have variable part \(m\), with exponent one. The coefficients do not need to match.
Q22 Practice 2 of 5 · Build
From the list t, 2t², 7t², and −3, select all terms that are like −5t². Explain your choices.
Answer
2t² and 7t².
- Like terms must have the same variable raised to the same power. The target term −5t² and both selected terms have variable part t².
- The coefficients −5, 2, and 7 may differ in value and sign; this does not prevent the terms from being like terms.
- The term t has exponent 1, not 2, and −3 is a constant. Neither is like −5t².
Q23 Practice 3 of 5 · Apply
Group the like terms in \(4a-7b+6+9a+2b-1\), without combining them yet.
Answer
\((4a+9a)+(-7b+2b)+(6-1)\).
- Make an \(a\)-group, a \(b\)-group, and a constant group. Preserve every sign.
Q24 Practice 4 of 5 · Reason
Are \(3ab\) and \(-8ba\) like terms? What about \(3ab\) and \(3a\)?
Answer
The first pair is like; the second pair is unlike.
- Multiplication can be reordered, so \(ba=ab\).
- The term \(3a\) lacks the factor \(b\), so its variable part differs.
Q25 Practice 5 of 5 · Challenge
Among \(2x^2y\), \(-6xy^2\), and \(4yx^2\), identify the like terms. Explain why all three cannot be combined.
Answer
\(2x^2y\) and \(4yx^2\) are like terms.
- Reorder \(yx^2\) as \(x^2y\).
- The term \(-6xy^2\) squares \(y\), not \(x\), so it has a different variable part.
Combining like terms
Add or subtract the coefficients of like terms and keep their variable part unchanged. Combining terms is addition, not multiplication, so exponents do not increase. Once all like groups have been combined, unlike groups remain separate.
Try it yourself: five questions, increasing difficulty
Q26 Practice 1 of 5 · Foundation
Simplify \(9x-4x\).
Answer
\(5x\).
- Subtract coefficients: \((9-4)x=5x\).
Q27 Practice 2 of 5 · Build
Simplify \(6a+7+2a-3\).
Answer
\(8a+4\).
- Combine \(6a+2a=8a\) and \(7-3=4\).
Q28 Practice 3 of 5 · Apply
Simplify \(5m-2n-8m+7n\).
Answer
\(-3m+5n\).
- The \(m\)-coefficient is \(5-8=-3\); the \(n\)-coefficient is \(-2+7=5\).
Q29 Practice 4 of 5 · Reason
Simplify \(4x^2+6x-x^2-9x+5\).
Answer
\(3x^2-3x+5\).
- Combine the square terms: \((4-1)x^2=3x^2\).
- Combine the first-power terms: \((6-9)x=-3x\). The constant stays five.
Q30 Practice 5 of 5 · Challenge
Simplify \(\frac12p+\frac34q-\frac32p+\frac14q+6-9\).
Answer
\(-p+q-3\).
- The \(p\)-coefficient is \(1/2-3/2=-1\). The \(q\)-coefficient is \(3/4+1/4=1\).
- The constants give \(6-9=-3\).
Using the distributive property
Multiply every term inside parentheses by the factor outside. An outside negative multiplies each sign as well as each number. After expanding, combine any like terms across the whole expression.
Try it yourself: five questions, increasing difficulty
Q31 Practice 1 of 5 · Foundation
Expand \(4(x+6)\).
Answer
\(4x+24\).
- Multiply both terms by four: \(4x+4(6)\).
Q32 Practice 2 of 5 · Build
Expand \(-2(3y+5)\).
Answer
\(-6y-10\).
- Multiply \(-2\) by \(3y\) and by \(5\). Both products are negative.
Q33 Practice 3 of 5 · Apply
Simplify \(-(2a-7)+3a\).
Answer
\(a+7\).
- The leading minus means multiply by \(-1\): \(-2a+7+3a\).
- Combine the \(a\)-terms to get \(a+7\).
Q34 Practice 4 of 5 · Reason
Simplify \(3(2x-5)-2(x+4)\).
Answer
\(4x-23\).
- Expand both products: \(6x-15-2x-8\).
- Combine to get \(4x-23\).
Q35 Practice 5 of 5 · Challenge
Simplify \(-\frac12(6p-8q+10)+2(p-q)\).
Answer
\(-p+2q-5\).
- The first product is \(-3p+4q-5\); the second is \(2p-2q\).
- Combine each variable group separately.
Simplifying before evaluating
Simplifying writes an equivalent expression that works for every allowed input. Evaluating finds its value for particular inputs. Distribute, combine like terms, and only then substitute when that makes the arithmetic easier; substituting first should give the same result.
Try it yourself: five questions, increasing difficulty
Q36 Practice 1 of 5 · Foundation
Simplify \(3x+2x+4\), then evaluate at \(x=3\).
Answer
\(5x+4\); value \(19\).
- Combine \(3x+2x\) to get \(5x\).
- Then \(5(3)+4=19\).
Q37 Practice 2 of 5 · Build
Simplify \(3(y+2)+2y\), then evaluate at \(y=4\).
Answer
\(5y+6\); value \(26\).
- Expand to \(3y+6+2y\), then combine.
- Substitute four: \(5(4)+6=26\).
Q38 Practice 3 of 5 · Apply
Simplify \(4(a-3)-2a+5\), then evaluate at \(a=-2\).
Answer
\(2a-7\); value \(-11\).
- \(4a-12-2a+5=2a-7\).
- Then \(2(-2)-7=-11\).
Q39 Practice 4 of 5 · Reason
Simplify \(2(3m-n)-3(m+2n)+4\), then evaluate at \(m=2,\ n=-1\).
Answer
\(3m-8n+4\); value \(18\).
- Expand to \(6m-2n-3m-6n+4\). Combine like terms to get \(3m-8n+4\).
- Substitute: \(6-8(-1)+4=6+8+4=18\).
Q40 Practice 5 of 5 · Challenge
Simplify \(5-2[3x-(x-4)]+x\), then evaluate at \(x=-3\).
Answer
\(-3x-3\); value \(6\).
- Inside brackets, \(3x-(x-4)=2x+4\).
- Now the expression is \(5-2(2x+4)+x\). Distribute to get \(5-4x-8+x\), then combine to obtain \(-3x-3\).
- At \(-3\), the value is \(9-3=6\).
Expressions versus equations
An expression names a value; an equation states that two expressions are equal. You evaluate an expression at a given input, but solve an equation to find the inputs that make it true. An equation can also be an identity, true for every allowed input.
Try it yourself: five questions, increasing difficulty
Q41 Practice 1 of 5 · Foundation
Classify \(5n-8\) and \(5n-8=17\) as an expression or an equation.
Answer
\(5n-8\) is an expression; \(5n-8=17\) is an equation.
- Only the second statement asserts an equality between two expressions.
Q42 Practice 2 of 5 · Build
Evaluate \(7t+2\) at \(t=3\). Are you finding a value or solving for an unknown?
Answer
\(23\); you are finding a value.
- The input is already supplied. Calculate \(7(3)+2=23\).
Q43 Practice 3 of 5 · Apply
Does \(x=5\) make \(4x-7=13\) true? Explain using substitution.
Answer
Yes.
- Substitute five on the left: \(4(5)-7=20-7=13\), which equals the right side.
Q44 Practice 4 of 5 · Reason
Solve \(6y+5=29\), then check the solution.
Answer
\(y=4\).
- Subtract five from both sides: \(6y=24\). Divide both sides by six.
- Check: \(6(4)+5=29\).
Q45 Practice 5 of 5 · Challenge
Is \(2(x+5)+x=3x+10\) true for one value of \(x\), no values, or all values? Explain without guessing.
Answer
It is true for all real values of \(x\).
- Simplify the left side to \(2x+10+x=3x+10\).
- Both sides are the same expression, so the equation is an identity, not an equation with just one solution.
Writing and evaluating expressions from stories
Define what your variable measures, then translate the story. Separate a one-time charge from a per-item charge. Write an expression, simplify it if useful, and substitute the requested value. Finish with units and an interpretation that fits the story.
Try it yourself: five questions, increasing difficulty
Q46 Practice 1 of 5 · Foundation
A ribbon costs $3 per meter. Write the cost of \(m\) meters, then find the cost of 7 meters.
Answer
\(3m\) dollars; $21.
- Multiply the cost per meter by the number of meters: \(3(7)=21\).
Q47 Practice 2 of 5 · Build
A skating rink charges $8 admission plus $5 per hour of skate rental. Write the cost of \(h\) hours and evaluate it for 3 hours.
Answer
\(8+5h\) dollars; $23.
- The admission fee is paid once. Rental for three hours costs \(5(3)=15\).
- The total is \(8+15=23\).
Q48 Practice 3 of 5 · Apply
One club makes \(4k+3\) cards and another makes \(2k+7\) cards. Write a simplified expression for their combined total, then evaluate at \(k=5\).
Answer
\(6k+10\) cards; 40 cards.
- Add the two expressions: \(4k+3+2k+7=6k+10\).
- At five, \(6(5)+10=40\).
Q49 Practice 4 of 5 · Reason
Three craft kits each contain \(2n+4\) beads. Five beads from the combined supply are set aside. Simplify the usable total and evaluate at \(n=6\).
Answer
\(6n+7\) beads; 43 beads.
- Write \(3(2n+4)-5\), then distribute: \(6n+12-5=6n+7\).
- At six, \(36+7=43\).
Q50 Practice 5 of 5 · Challenge
A robot’s battery-change model is \(4t-3(t+2)\) percentage points after \(t\) minutes. Simplify and evaluate at \(t=4\). What does the sign of the answer mean?
Answer
\(t-6\); \(-2\) percentage points, a decrease of 2 percentage points.
- Distribute the subtraction: \(4t-3t-6=t-6\).
- At four, the result is \(-2\). This is a change in charge, not a negative battery level.
Ten final word problems
Define the variable and write an expression before calculating. The first problems use one operation; later problems require distribution, subtraction of a whole expression, negative values, and an equation. Give a complete sentence interpreting your result.
W1 Buying pencils
Pencils cost $2 each. Maya buys \(p\) pencils. Write an expression for the cost, then evaluate it when \(p=9\).
Answer
\(2p\) dollars; $18.
- Each pencil contributes two dollars, so multiply \(2\) by \(p\).
- For nine pencils, \(2(9)=18\).
W2 A bicycle rental
A bicycle rental costs $6 to unlock the bike and $4 per hour. Write the cost for \(h\) hours and find the cost of 5 hours.
Answer
\(6+4h\) dollars; $26.
- The unlock fee is paid once, not once per hour.
- \(6+4(5)=6+20=26\).
W3 Temperature changes
At dawn the temperature is \(-3^\circ\mathrm C\). It rises by \(2^\circ\mathrm C\) each hour for \(h\) hours. Write the modeled temperature and evaluate it for \(h=4\).
Answer
\(-3+2h\); \(5^\circ\mathrm C\).
- Start with the initial temperature and add the rise.
- \(-3+2(4)=-3+8=5\).
W4 Combining fundraiser sales
A morning fundraiser sells \(3n+5\) tickets and the afternoon fundraiser sells \(4n-2\). Write a simplified expression for total ticket sales and evaluate at \(n=6\).
Answer
\(7n+3\) tickets; 45 tickets.
- Add the totals: \(3n+5+4n-2=7n+3\).
- At six, \(7(6)+3=45\).
W5 A garden border
A rectangular garden has length \(2x+3\) meters and width \(x+1\) meters. Write and simplify its perimeter, then evaluate at \(x=4\).
Answer
\(6x+8\) meters; 32 meters.
- A rectangle has two lengths and two widths: \(2(2x+3)+2(x+1)\).
- Distribute and combine: \(4x+6+2x+2=6x+8\). At four, \(24+8=32\).
W6 Identical supply boxes
A teacher buys 4 supply boxes. Each contains \(3b+2\) pencils, and 6 pencils from the combined supply are saved for another class. Write a simplified expression for the pencils available now, then evaluate at \(b=3\).
Answer
\(12b+2\) pencils; 38 pencils.
- The available total is \(4(3b+2)-6\).
- Expand to \(12b+8-6=12b+2\). At three, \(36+2=38\).
W7 Two workshop groups
Each student in Group A pays $7 and each student in Group B pays $5. The groups have \(2k+1\) and \(k+3\) students respectively. Write a simplified expression for the combined fees, then evaluate at \(k=4\).
Answer
\(19k+22\) dollars; $98.
- Write \(7(2k+1)+5(k+3)\).
- Distribute to get \(14k+7+5k+15=19k+22\). At four, \(76+22=98\).
W8 Comparing printing plans
Printer A charges \(12+2p\) dollars for \(p\) posters. Printer B charges \(5+3p\) dollars. Write and simplify “A’s cost minus B’s cost.” Evaluate it for 9 posters and explain which printer is cheaper.
Answer
\(7-p\); \(-2\) dollars at \(p=9\). Printer A is $2 cheaper.
- Subtract the whole second cost: \((12+2p)-(5+3p)\). Remove parentheses to get \(12+2p-5-3p\), then combine to obtain \(7-p\).
- At nine the difference is \(-2\), so A costs less. Directly, A costs $30 and B costs $32.
W9 A game-score rule
A game awards a score modeled by \(3r^2+2r-5\), where \(r\) is a signed round result. Find the score when \(r=-2\). A player claims the square contributes \(-12\); explain the error.
Answer
The score is \(3\) points. The square term contributes \(12\), not \(-12\).
- Substitute with parentheses: \(3(-2)^2+2(-2)-5\).
- Since \((-2)^2=4\), the score is \(12-4-5=3\). The minus sign is part of the number being squared.
W10 A school-fair budget
A club collects $8 from each of \(n\) students and receives a $25 donation. It buys 3 supply packs costing \(2n+4\) dollars each and spends another $7 on signs. Write a simplified expression for the money left, evaluate it for \(n=14\), and find how many students are needed to have exactly $40 left.
Answer
\(2n+6\) dollars left; $34 for 14 students; 17 students for $40 left.
- Income is \(8n+25\). Total expense is \(3(2n+4)+7=6n+19\).
- Subtract all expenses: \(8n+25-(6n+19)=2n+6\). At fourteen, \(28+6=34\).
- For exactly forty dollars, solve \(2n+6=40\). Then \(2n=34\), so \(n=17\). Check: income $161 minus expenses $121 leaves $40.
Quick questions and answers
What does it mean to evaluate an algebraic expression?
Replace every variable with its given value, then calculate using the order of operations. Evaluation produces a value for those particular inputs.
What are like terms?
Like terms have the same variable part with the same exponents. Their coefficients may be different. Constants are like terms with other constants.
How do you combine like terms?
Add or subtract their coefficients and keep the variable part unchanged. Combining terms does not increase exponents.
How do you substitute negative values safely?
Put each negative value in parentheses. Distinguish squaring the whole negative number from applying a minus sign after a square.
How does the distributive property work?
Multiply every term inside parentheses by the outside factor, including its sign, then combine like terms if possible.
What is the difference between an expression and an equation?
An expression represents a value. An equation asserts that two expressions are equal. Evaluate an expression at supplied inputs; solve an equation to find the inputs that make it true.
What practice is included in Pre-Algebra Class 12?
Ten posters appear in sequence, each followed by five original questions that increase in difficulty. Ten cumulative word problems follow, giving sixty questions with hidden worked solutions.