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Student "Basics" Package

 

The new Student Basics package helps to explore the foundations of higher math, making it possible to provide step-by-step breakdowns for expanding and simplifying mathematical expressions, such as simplifying fractions, expanding products of polynomials, or solving linear equations. All the steps to the solution are shown and documented, so that a student can easily understand what is happening at each stage of the solution. Students can use this package to understand where results are coming from and learn how to solve these problems on their own.

 

Here are 101 interesting examples showing the steps involved to solve or expand:

 

withStudent:-Basics:

 

LinearSolveStepsx + 3 = 8, x;

x+3=8x=83subtract from both sidesx=5add terms

(1)

LinearSolveSteps5x2 = 13, x;

5x2=135x=13+2subtract from both sidesx=13+25divide both sidesx=155add termsx=3reduce fraction by gcd

(2)

LinearSolveSteps4x + 5 = x - 4, x, implicitmultiply;

4x+5=x44xx=−45subtract from both sides3x=−45add terms3x=−9add termsx=−93divide both sidesx=−3reduce fraction by gcd

(3)

LinearSolveSteps3(n - 1.8) = 2n + 1, n, implicitmultiply;

3n1.8=2n+13n+3−1.8=2n+1distributive multiply3n5.4=2n+1multiply constants3n2n=1+5.4subtract from both sidesn=1+5.4add termsn=6.4add terms

(4)

LinearSolveSteps7y + 5 - 3y + 1 = 2y + 2 , y, implicitmultiply;

7y+53y+1=2y+27y3y2y=251subtract from both sides2y=251add terms2y=−4add termsy=−42divide both sidesy=−2reduce fraction by gcd

(5)

LinearSolveSteps2x + 4 = 10, x, implicitmultiply;

2x+4=102x=104subtract from both sidesx=1042divide both sidesx=62add termsx=3reduce fraction by gcd

(6)

LinearSolveSteps3x - 4 = -10, x, implicitmultiply; 

3x4=−103x=−10+4subtract from both sidesx=−10+43divide both sidesx=−63add termsx=−2reduce fraction by gcd

(7)

LinearSolveSteps4x - 4y = 8, x, implicitmultiply; 

4x4y=84x=8+4ysubtract from both sidesx=8+4y4divide both sidesx=42+y4factorx=2+ydivide

(8)

LinearSolveStepsx + 3^2 = 12 , x;

x+32=12x=1232subtract from both sidesx=129evaluate powerx=3add terms

(9)

LinearSolveStepsx + y2 = 12, x;

y2+x=12x=12y2subtract from both sidesx=y2+12reorder terms

(10)

LinearSolveSteps x2 + y24 = 14 x2  2x + 14, x 

x24+y24=x24+−2x+14x24x24−2x=14y24subtract from both sidesx24x24+2x=14y24distribute negation2x=14y24add termsx=14y242divide both sides

(11)

LinearSolveSteps4(8-3x) = 32 - 8(x+2), x, implicitmultiply;

483x=328x+24·8+43x=328x+2distributive multiply32+43x=328x+2multiply constants32+−12x=328x+2multiply constants−12x+8x+2=3232subtract from both sides12x+8x+8·2=3232distributive multiply12x+8x+16=3232multiply constants4x+16=3232add terms4x+16=0add terms4x=−16subtract from both sidesx=−16−4divide both sidesx=4reduce fraction by gcd

(12)

LinearSolveSteps2x/3 = -18, x, implicitmultiply;

2x3=−182x=3−18multiply rhs by denominator of lhsx=3−182divide both sidesx=−542multiply constantsx=−27reduce fraction by gcd

(13)

LinearSolveSteps2/(3x) = -18, x, implicitmultiply;

23x=−18123x=118reciprocal of both sidesx=11832divide both sidesx=11823rewrite division as multiplication by reciprocalx=127multiply fraction and reduce by gcd

(14)

LinearSolveSteps(2x)/(3x^2) = -18, x, implicitmultiply;

2x3x2=−1823x=−18divide out common terms3x2=118reciprocal of both sidesx=11832divide both sidesx=11823rewrite division as multiplication by reciprocalx=127multiply fraction and reduce by gcd

(15)

LinearSolveSteps2/(3x) = -18, x, implicitmultiply;

23x=−18123x=118reciprocal of both sidesx=11832divide both sidesx=11823rewrite division as multiplication by reciprocalx=127multiply fraction and reduce by gcd

(16)

LinearSolveSteps(2x)/(3x^2) = -18, x, implicitmultiply; 

2x3x2=−1823x=−18divide out common terms3x2=118reciprocal of both sidesx=11832divide both sidesx=11823rewrite division as multiplication by reciprocalx=127multiply fraction and reduce by gcd

(17)

LinearSolveStepsx/3-5/12 = 3/4+1/2x, x, implicitmultiply;

x3512=34+12xx312x=34+512subtract from both sidesx3x2=34+512multiply fractionx6=34+512add termsx6=76add termsx=676multiply rhs by denominator of lhsx=426multiply fractionx=7reduce fraction by gcdx=−7negate both sides

(18)

LinearSolveSteps10-3x = 7, x, implicitmultiply; 

103x=73x=710subtract from both sides3x=−3add termsx=−3−3divide both sidesx=1divide out common terms

(19)

LinearSolveSteps2(x+5)-7 = 3(x-2), x, implicitmultiply;

2x+57=3x22x+57=3x+3−2distributive multiply2x+57=3x6multiply constants2x+53x=−6+7subtract from both sides2x+2·53x=−6+7distributive multiply2x+103x=−6+7multiply constantsx+10=−6+7add termsx+10=1add termsx=110subtract from both sidesx=−9add termsx=9negate both sides

(20)

LinearSolveSteps-x = 1+2, x;  

x=1+2x=3add termsx=−3negate both sides

(21)

LinearSolveSteps-x = 1+y, x; 

x=1+yx=1+ynegate both sidesx=1ydistribute negation

(22)

LinearSolveSteps5x/4+1/2 = 2x-1/2, x, implicitmultiply;  

5x4+12=2x125x42x=1212subtract from both sides3x4=1212add terms3x4=−1add terms3x=4−1multiply rhs by denominator of lhs3x=−4multiply constantsx=−4−3divide both sidesx=43reduce fraction by gcd

(23)

LinearSolveSteps.35y - .2 = .15y + .1, y, implicitmultiply;

0.35y0.2=0.15y+0.10.35y0.15y=0.1+0.2subtract from both sides0.20y=0.1+0.2add terms0.20y=0.3add termsy=0.30.20divide both sidesy=1.500000000divide constants

(24)

LinearSolveSteps4x-1 = 4(x+3), x, implicitmultiply;

4x1=4x+34x1=4x+4·3distributive multiply4x1=4x+12multiply constants4x4x=12+1subtract from both sides0=12+1add terms0=13add terms0=13no solution

(25)

LinearSolveSteps5x+10 = 5(x+2), x, implicitmultiply;

5x+10=5x+25x+10=5x+5·2distributive multiply5x+10=5x+10multiply constants5x5x=1010subtract from both sides0=1010add terms0=0add terms0=0infinite number of solutions

(26)

LinearSolveStepsxy + 6x = 1, x, implicitmultiply;

xy+6x=1x6+y=1factorx=16+ydivide both sides

(27)

LinearSolveSteps(x+1)/(2*y*z) = 4*y^2/z + 3*x/y, x; 

x+12yz=4y2z+3xyx+12yz3xy=4y2zsubtract from both sidesyx+12yzy+2yz3x2yzy=4y2zfind common denominatoryx+1+2yz3x2yzy=4y2zsum over common denominatoryx+y·1+2yz3x2yzy=4y2zdistributive multiplyxy+y+−6yzx2yzy=4y2zmultiply constants6xyz+xy+y2yzy=4y2zreorder termsy6xz+x+1y·2yz=4y2zfactor6xz+x+12yz=4y2zdivide6xz+x+1=2yz4y2zmultiply rhs by denominator of lhs6xz+x=2yz4y2z1subtract from both sides6xz+x=8y3zz1multiply fraction6xz+x=8y31dividex16z=8y31factorx=8y3116zdivide both sides

(28)

LinearSolveSteps1/x = 3/4, x;  

1x=34x=43reciprocal of both sides

(29)

LinearSolveSteps1/x = 4, x;  

1x=4x=14reciprocal of both sides

(30)

LinearSolveSteps1/x - 1/2 = 3/4 - 2/x, x;  

1x12=342x1x+2x=34+12subtract from both sides3x=34+12add terms3x=54add termsx3=45reciprocal of both sidesx=4513divide both sidesx=4531rewrite division as multiplication by reciprocalx=125multiply fraction and reduce by gcd

(31)

LinearSolveSteps3(n - 1.8) + 2(n-1) = 2(n + 1) - 3(n-2), n, implicitmultiply;

3n1.8+2n1=2n+13n23n1.8+2n12n+1+3n2=0subtract from both sides3n+3−1.8+2n12n+1+3n2=0distributive multiply3n5.4+2n12n+1+3n2=0multiply constants3n5.4+2n+2−12n+1+3n2=0distributive multiply3n5.4+2n22n+1+3n2=0multiply constants3n5.4+2n22n+2·1+3n2=0distributive multiply3n5.4+2n22n+2+3n2=0multiply constants3n5.4+2n22n+2+3n+3−2=0distributive multiply3n5.4+2n22n+2+3n6=0multiply constants6n15.4=0add terms6n=15.4subtract from both sidesn=15.46divide both sidesn=2.566666667divide constants

(32)

LinearSolveSteps3(n - 1.8) + n*(2-1) = 2n + 1, n, implicitmultiply;

3n1.8+n21=2n+13n1.8+n212n=1subtract from both sides3n+3−1.8+n212n=1distributive multiply3n5.4+n212n=1multiply constants3n5.4+n·12n=1add terms2n5.4=1add terms2n=1+5.4subtract from both sides2n=6.4add termsn=6.42divide both sidesn=3.200000000divide constants

(33)

LinearSolveSteps5x/4+1/2 = 2x-1/2, x, implicitmultiply; 

5x4+12=2x125x42x=1212subtract from both sides3x4=1212add terms3x4=−1add terms3x=4−1multiply rhs by denominator of lhs3x=−4multiply constantsx=−4−3divide both sidesx=43reduce fraction by gcd

(34)

LinearSolveStepsx*(1-6*z) = 8*y-1, x;

x16z=8y1x=8y116zdivide both sides

(35)

LinearSolveSteps3*(x-6*z) = 8*y-1, x;  

3x6z=8y13x+36z=8y1distributive multiply3x+−18z=8y1multiply constants3x=8y1−18zsubtract from both sides3x=8y1+18zdistribute negationx=8y1+18z3divide both sides

(36)

LinearSolveSteps10-3x = 7+2x, x, implicitmultiply;

103x=7+2x3x2x=710subtract from both sides5x=710add terms5x=−3add termsx=−3−5divide both sidesx=35reduce fraction by gcd

(37)

LinearSolveSteps10-3x = 7, x, implicitmultiply; 

103x=73x=710subtract from both sides3x=−3add termsx=−3−3divide both sidesx=1divide out common terms

(38)

LinearSolveSteps103x = 7+3xy/4z, x; 

10+−3x=7+3x+14yz−3x7+3x+14yz=−10subtract from both sides3x7+3x+y4z=−10multiply fractionz3xz+7+3x+y4z=−10find common denominatorz3x7+3x+y4z=−10sum over common denominator3xz73x+y4z=−10distribute negation3xz73x+y4=z−10multiply rhs by denominator of lhs3xz3x=z−10+7y4subtract from both sides3xz3x=710zy4reorder termsx−33z=710zy4factorx=710zy4−33zdivide both sides

(39)

LinearSolveSteps1x+2=1,x;

x+2−1=1x+2=1reciprocal of both sidesx=12subtract from both sidesx=−1add terms

(40)

LinearSolveStepsxx+2=1,x;

xx+2=1x=x+2·1multiply rhs by denominator of lhsx=x+2multiply by 1xx=2subtract from both sides0=2add terms0=2no solution

(41)

ExpandSteps(3*a)*(4*a-y+42);

3a4ay+42=3a·4a+3ay+3a·42distributive multiply=12aa+3ay+3a·42multiply constants=12a2+3ay+3a·42multiply terms to exponential form=12a2+−3ay+3a·42multiply constants=12a23ay+126amultiply constants

(42)

ExpandSteps(3*a)*(4*a-y+42);    

3a4ay+42=3a·4a+3ay+3a·42distributive multiply=12aa+3ay+3a·42multiply constants=12a2+3ay+3a·42multiply terms to exponential form=12a2+−3ay+3a·42multiply constants=12a23ay+126amultiply constants

(43)

ExpandSteps(x^2)*(x^3);

x2x3=x5add exponents with common base

(44)

ExpandSteps(x^2*y/(x*y)); 

x2yxy=x2xdivide out common terms=xdivide

(45)

ExpandSteps(2*x^2*y/(4*x*y));

2x2y4xy=2x24xdivide out common terms=2x4divide out common terms=x2reduce fraction by gcd

(46)

ExpandSteps(2.1*x)/4.3; 

2.1x4.3=0.4883720930xdivide constants

(47)

ExpandSteps(2.1*x^2*y/(4.3*x*y));

2.1x2y4.3xy=2.1x24.3xdivide out common terms=2.100000000x4.3divide out common terms=0.4883720930xdivide constants

(48)

ExpandSteps(x^2*y+y^2⋅x)/(x+y);

x2y+y2xx+y=x+yxyx+yfactor=xydivide

(49)

ExpandSteps(-y)^2;

y2=−12y2distribute exponent to individual terms=1y2evaluate power

(50)

ExpandSteps(-y^2)+y^2;

y2+y2=0add terms

(51)

ExpandSteps(x^2-y^2)/(x+y);

x2y2x+y=x+yxyx+yfactor=xydivide

(52)

ExpandSteps2*(-y^2);

2y2=−2y2multiply constants

(53)

ExpandSteps2*(x^2-y^2);

2x2y2=2x2+2y2distributive multiply=2x2+−2y2multiply constants

(54)

ExpandSteps(2*(x^2-y^2))/(4*(x+y));

2x2y24x+y=2x2+2y24x+ydistributive multiply=2x2+−2y24x+ymultiply constants=2x22y24x+4ydistributive multiply=2x+2yxy2x+2y·2factor=xy2divide

(55)

ExpandSteps(2.1*(x^2-y^2))/4;

2.1x2y24=2.1x2+2.1y24distributive multiply=2.1x2+−2.1y24multiply constants=0.5250000000x20.5250000000y2divide constants

(56)

ExpandSteps(2.1*(x^2-y^2))/(4*(x+y));

2.1x2y24x+y=2.1x2+2.1y24x+ydistributive multiply=2.1x2+−2.1y24x+ymultiply constants=2.1x22.1y24x+4ydistributive multiply=x+y2.100000000x2.100000000yx+y·4.factor=2.100000000x2.100000000y4.divide=0.5250000000x0.5250000000ydivide constants

(57)

ExpandSteps(x^2/z)*(z^3/x^2);

x2zz3x2=x2z3zx2multiply fraction=x2z2x2divide out common terms=z2divide out common terms

(58)

ExpandSteps(17*x^4*y^2/(64*z^5)) * (24*y*z^2/(85*x^2));

17x4y264z524yz285x2=408x4y3z25440z5x2multiply fraction=408x4y35440x2z3divide out common terms=408x2y35440z3divide out common terms=3x2y340z3reduce fraction by gcd

(59)

ExpandSteps3^2;

32=9evaluate power

(60)

ExpandSteps`%+``%+`9*a^2,6*a*b,`%+`6*a*b,4*b^2;

9a2+6ab+6ab+4b2=9a2+12ab+4b2add terms

(61)

ExpandSteps(3*a+2*b)^2;

3a+2b2=3a+2b3a+2brewrite exponentiation as multiplication=3a3a+2b+2b3a+2bdistributive multiply=3a·3a+3a·2b+2b3a+2bdistributive multiply=9aa+3a·2b+2b3a+2bmultiply constants=9a2+3a·2b+2b3a+2bmultiply terms to exponential form=9a2+6ab+2b3a+2bmultiply constants=9a2+6ab+2b·3a+2b·2bdistributive multiply=9a2+6ab+6ba+2b·2bmultiply constants=9a2+6ab+6ab+4bbmultiply constants=9a2+6ab+6ab+4b2multiply terms to exponential form=9a2+12ab+4b2add terms

(62)

ExpandSteps(3a+2b)*(4a-y+42), implicitmultiply;

3a+2b4ay+42=3a4ay+42+2b4ay+42distributive multiply=3a·4a+3ay+3a·42+2b4ay+42distributive multiply=12aa+3ay+3a·42+2b4ay+42multiply constants=12a2+3ay+3a·42+2b4ay+42multiply terms to exponential form=12a2+−3ay+3a·42+2b4ay+42multiply constants=12a23ay+126a+2b4ay+42multiply constants=12a23ay+126a+2b·4a+2by+2b·42distributive multiply=12a23ay+126a+8ba+2by+2b·42multiply constants=12a23ay+126a+8ab+−2by+2b·42multiply constants=12a23ay+126a+8ab2by+84bmultiply constants=12a2+8ab3ay2by+126a+84breorder terms

(63)

ExpandSteps3*3;

3·3=9multiply constants

(64)

ExpandSteps1*2*3*4*5*6*7*8*9;

1·2·3·4·5·6·7·8·9=362880multiply constants

(65)

ExpandSteps1+1;

1+1=2add terms

(66)

ExpandSteps0^x;

0x=0evaluate power

(67)

ExpandStepsx^0;

x0=1x^0 = 1

(68)

ExpandSteps5^0;

50=1x^0 = 1

(69)

ExpandSteps(a*b)^3;

ab3=a3b3distribute exponent to individual terms

(70)

ExpandStepsa^3*a^2;

a3a2=a5add exponents with common base

(71)

ExpandSteps(a+b)^2;

a+b2=a+ba+brewrite exponentiation as multiplication=aa+b+ba+bdistributive multiply=aa+ab+ba+bdistributive multiply=a2+ab+ba+bmultiply terms to exponential form=a2+ab+ba+bbdistributive multiply=a2+ab+ab+b2multiply terms to exponential form=a2+2ab+b2add terms

(72)

ExpandStepsa+b5;

a+b5=a+ba+ba+ba+ba+brewrite exponentiation as multiplication=aa+b+ba+ba+ba+ba+bdistributive multiply=aa+ab+ba+ba+ba+ba+bdistributive multiply=a2+ab+ba+ba+ba+ba+bmultiply terms to exponential form=a2+ab+ba+bba+ba+ba+bdistributive multiply=a2+ab+ab+b2a+ba+ba+bmultiply terms to exponential form=a2+2ab+b2a+ba+ba+badd terms=a2+2ab+b2a+a2+2ab+b2ba+ba+bdistributive multiply=aa2+a·2ab+ab2+a2+2ab+b2ba+ba+bdistributive multiply=a3+a·2ab+ab2+a2+2ab+b2ba+ba+badd exponents with common base=a3+2a2b+ab2+a2+2ab+b2ba+ba+bmultiply terms to exponential form=a3+2a2b+ab2+ba2+b·2ab+bb2a+ba+bdistributive multiply=a3+2a2b+ab2+a2b+2b2a+bb2a+ba+bmultiply terms to exponential form=a3+2a2b+ab2+a2b+2ab2+b3a+ba+badd exponents with common base=a3+3a2b+3ab2+b3a+ba+badd terms=a3+3a2b+3ab2+b3a+a3+3a2b+3ab2+b3ba+bdistributive multiply=aa3+a·3a2b+a·3ab2+ab3+a3+3a2b+3ab2+b3ba+bdistributive multiply=a4+a·3a2b+a·3ab2+ab3+a3+3a2b+3ab2+b3ba+badd exponents with common base=a4+3a3b+a·3ab2+ab3+a3+3a2b+3ab2+b3ba+badd exponents with common base=a4+3a3b+3a2b2+ab3+a3+3a2b+3ab2+b3ba+bmultiply terms to exponential form=a4+3a3b+3a2b2+ab3+ba3+b·3a2b+b·3ab2+bb3a+bdistributive multiply=a4+3a3b+3a2b2+ab3+a3b+3b2a2+b·3ab2+bb3a+bmultiply terms to exponential form=a4+3a3b+3a2b2+ab3+a3b+3a2b2+3b3a+bb3a+badd exponents with common base=a4+3a3b+3a2b2+ab3+a3b+3a2b2+3ab3+b4a+badd exponents with common base=a4+4a3b+6a2b2+4ab3+b4a+badd terms=a4+4a3b+6a2b2+4ab3+b4a+a4+4a3b+6a2b2+4ab3+b4bdistributive multiply=aa4+a·4a3b+a·6a2b2+a·4ab3+ab4+a4+4a3b+6a2b2+4ab3+b4bdistributive multiply=a5+a·4a3b+a·6a2b2+a·4ab3+ab4+a4+4a3b+6a2b2+4ab3+b4badd exponents with common base=a5+4a4b+a·6a2b2+a·4ab3+ab4+a4+4a3b+6a2b2+4ab3+b4badd exponents with common base=a5+4a4b+6a3b2+a·4ab3+ab4+a4+4a3b+6a2b2+4ab3+b4badd exponents with common base=a5+4a4b+6a3b2+4a2b3+ab4+a4+4a3b+6a2b2+4ab3+b4bmultiply terms to exponential form=a5+4a4b+6a3b2+4a2b3+ab4+ba4+b·4a3b+b·6a2b2+b·4ab3+bb4distributive multiply=a5+4a4b+6a3b2+4a2b3+ab4+a4b+4b2a3+b·6a2b2+b·4ab3+bb4multiply terms to exponential form=a5+4a4b+6a3b2+4a2b3+ab4+a4b+4a3b2+6b3a2+b·4ab3+bb4add exponents with common base=a5+4a4b+6a3b2+4a2b3+ab4+a4b+4a3b2+6a2b3+4b4a+bb4add exponents with common base=a5+4a4b+6a3b2+4a2b3+ab4+a4b+4a3b2+6a2b3+4ab4+b5add exponents with common base=a5+5a4b+10a3b2+10a2b3+5ab4+b5add terms

(73)

Note that this could be expanded but the system chooses not to as the output would be excessively large (the cut-off is an exponent  100)

  ExpandStepsa+b1000; 

a+b1000

(74)

ExpandSteps(a+b)^(1/2) ⋅ (a+b)^(3/2);

a+b12a+b32=a+b2add exponents with common base=a+ba+brewrite exponentiation as multiplication=aa+b+ba+bdistributive multiply=aa+ab+ba+bdistributive multiply=a2+ab+ba+bmultiply terms to exponential form=a2+ab+ba+bbdistributive multiply=a2+ab+ab+b2multiply terms to exponential form=a2+2ab+b2add terms

(75)

ExpandSteps(1+I)^1.5;

1+I1.5=1+I1.5add terms=0.6435942529+1.553773974·Ievaluate power

(76)

ExpandStepsa/(2*b/a);

a2ba=aa2brewrite division as multiplication by reciprocal=a22bmultiply fraction and reduce by gcd

(77)

ExpandStepsa/(2*a);

a2a=12divide out common terms

(78)

ExpandSteps3a/(6a), implicitmultiply;

3a6a=36divide out common terms=12reduce fraction by gcd

(79)

ExpandSteps(3x sin(x))/x, implicitmultiply; 

3xsinxx=3sinxdivide out common terms

(80)

ExpandSteps3( sin(x) + y ), implicitmultiply;

3sinx+y=3sinx+3ydistributive multiply

(81)

ExpandSteps(3a+2b)*(4a-y^2+42), implicitmultiply;

3a+2b4ay2+42=3a+2by2+4a+42reorder terms=y2+4a+42·3a+y2+4a+42·2bdistributive multiply=3ay2+3a·4a+3a·42+y2+4a+42·2bdistributive multiply=−3ay2+3a·4a+3a·42+y2+4a+42·2bmultiply constants=3ay2+12aa+3a·42+y2+4a+42·2bmultiply constants=3ay2+12a2+3a·42+y2+4a+42·2bmultiply terms to exponential form=3ay2+12a2+126a+y2+4a+42·2bmultiply constants=3ay2+12a2+126a+2by2+2b·4a+2b·42distributive multiply=3ay2+12a2+126a+−2by2+2b·4a+2b·42multiply constants=3ay2+12a2+126a2by2+8ba+2b·42multiply constants=3ay2+12a2+126a2by2+8ab+84bmultiply constants=3ay22by2+12a2+8ab+126a+84breorder terms

(82)

ExpandSteps3(a^2-1)/(a+1), implicitmultiply;

3a21a+1</