Using a computer algebra system (CAS) to efficiently and correctly do Algebra

https://en.wikipedia.org/wiki/Computer_algebra_system
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https://www.wolfram.com/language/elementary-introduction/3rd-ed/preface.html
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https://www.wolframcloud.com/obj/online-courses/an-elementary-introduction-to-the-wolfram-language/01-starting-out-elementary-arithmetic.html
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https://www.wolframcloud.com/obj/online-courses/introduction-to-algebra/what-is-algebra.html
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I. Operations with Numbers - (+, -, x, ÷,
√
)

Enter the numerical expression you wish to compute, (using parentheses for grouping) and pressing shift/return (or shift/enter) to obtain the fully simplified result.
In[]:=
HoldForm[746+472]
Out[]=
746+472
In[]:=
ReleaseHold[HoldForm[746+472]]
Out[]=
1218
In[]:=
746+472
Out[]=
746+472
​
In[]:=
HoldForm[(-3/2)(16/19)]
Out[]=
-
3×16
2×19
In[]:=
HoldForm[3/14-14/9]
Out[]=
3
14
-
14
9
In[]:=
HoldForm[Sqrt[112]]
Out[]=
112
In[]:=
HoldForm[Sqrt[27/169]]
Out[]=
27
169
In[]:=
HoldForm[(5/2-9/4)/(5/6)]
Out[]=
5
2
-
9
4
5
6
Example #0: find the slope of the line that passes through the points (1, 1) and (-5, -8)
In[]:=
(-8-1)/(-5-1)
Out[]=
3
2

II. Evaluating Expressions

Example #1: Evaluate 4 x -7 when x = 11
In[]:=
4x-7/.x->11
Out[]=
37
This means replace x with 7 in the expression 4x -7 and simplify
Example #2: Evaluate 9
2
y
+ 6y - 5 when y = 2.
In[]:=
9y^2+6y-5/.y->2
Out[]=
43
Example #3: Evaluate 4w - 6x when w = 10 and x = 6.
In[]:=
4w-6x/.{w->10,x->6}
Out[]=
4

Extension/Application: generating a list of points

y
=
3
4
x +4 is the equation of a straight line with slope
3
4
and y-intercept (0,4)
The code below generates a list of values beginning at -5 and increasing by .1 up to 5
In[]:=
xvalues=Range[-5,5,.1]
Out[]=
{-5.,-4.9,-4.8,-4.7,-4.6,-4.5,-4.4,-4.3,-4.2,-4.1,-4.,-3.9,-3.8,-3.7,-3.6,-3.5,-3.4,-3.3,-3.2,-3.1,-3.,-2.9,-2.8,-2.7,-2.6,-2.5,-2.4,-2.3,-2.2,-2.1,-2.,-1.9,-1.8,-1.7,-1.6,-1.5,-1.4,-1.3,-1.2,-1.1,-1.,-0.9,-0.8,-0.7,-0.6,-0.5,-0.4,-0.3,-0.2,-0.1,2.77556×
-16
10
,0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1.,1.1,1.2,1.3,1.4,1.5,1.6,1.7,1.8,1.9,2.,2.1,2.2,2.3,2.4,2.5,2.6,2.7,2.8,2.9,3.,3.1,3.2,3.3,3.4,3.5,3.6,3.7,3.8,3.9,4.,4.1,4.2,4.3,4.4,4.5,4.6,4.7,4.8,4.9,5.}
Evaluates the function at each of the x-values
​
​
(
1
)
Below, the function Riffle creates a list alternating the x-values and the y-values and the function Partition separates them into ordered pairs
Point converts each of the ordered pairs into a graphics primitive and Graphics displays the result

III. Solving Equations and Inequalities

Part One: single variable linear equations

Use the built-in function Solve
Example #4: Solve for x. -4(2x + 2) - x - 5 = 23

Part Two: single variable linear inequalities

Use the built-in function Reduce
Example #6: solve the following inequality for w. Write your answer in simplest form. 8w - 2 ⩽ 6w - 10
We can use NumberLinePlot to see a plot of the solution

Part Three: use Solve and Simplify to transform a two variable linear equation into slope-intercept for (in other words, solve for y)

Example #7 : Put the following equation of a line into slope-intercept form. 5x + 6y = -36

IV. Solving systems of linear equations with Solve

Method: Solve[{list of equations}, {variables}]
Example #9: solve the following system of equations:
​y = 7x​
​x - 7y =-30
Checking the solution in the original equations

V. Graphing Linear inequalities

The function RegionPlot is used to plot inequalities. At a minimum,it requires an expression, a range for the horizontal variable and a range for the vertical variable. Note that is does not pro

Aside: Finding intercepts

To find the y-intercept we evaluate the expression when x = 0; to find the x-intercept we replace y with 0 and solve the resulting equation for x
The y-intercept is the point (0,7)

VI. Simplifying Expressions