Module 1 : Algebra

103 SOLVING EQUATIONS

S E T U P

In this project we will use the following command packages. Type and execute this line before begining the project below. If you re-enter the worksheet for this project, be sure to re-execute this statement before jumping to any point in the worksheet.

> restart;

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A. Exact Solutions : Solve

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In general, when we solve an equation, we usually obtain an exact solutions - which could be an integer, a fraction, or expressions with roots or in them. Using a calculator, we can convert exact solutions into decimal approximations of varying degrees of accuracy. Using Maple, we can solve many types of equations and inequalities and obtain exact solutions. Lets look at some of the different types of equations and how their solutions appear. In a later section of this module well look at decimal solutions.

EQUATIONS

Here are some examples of a linear and quadratic equations. equation

> solve( 4*x + 3 = 7);

> solve( 5* x^2 + 6*x =11 );

> solve( 2*x^2 -x-4);

> solve( x^4 + 3*x^2 -4);

> solve( 5*x^2 + 6*x = 11);

> solve( 2*x^2 - x - 4);

Not all equations are solvable algebraically.

> solve( x^5 - x + 1 );

Maple can also solve absolute value equations

> abs( 3*x -5) = 17; solve( %, x);

INEQALITIES

In the examples above, Maple returns one or more individual answers. However there are other types of problems, - for example those involving inequalities - where the answer involves intervals instead of individual numbers. Happily, Maple adjusts automatically to the type of problem involved. The same solve command will work on these problems. Maple expresses an interval as a real range.

> solve( 4*x +3 < 7);

> solve( 4*x +3 >=7 );

Here are some other types of inequalities.

-absolute value inequalities

> abs( 139 - 41*x ) > 73; solve(%);

- polynomial inequalities

> solve(x^2 + 3 >= 7);

-Higher degree polynomial inequalities

> poly := x^4 + 3*x^3 - 27*x^2 + 13*x + 42;

> solve( poly > 0,x);

A1. Find the solutions to these equations and inequalities

A. 3 - x - 4 = 0 B. - x - 4 = 0 C. 19x + 31 7
D. | 5x+ 2 | = 9 E. | 7 - 3x | 3 F. 4 + 3x 7
G. | x3 | + | x2 | = | x+13 | H. | |x+1| - |x-1| | < 2

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B. Equations with Multiple Unknowns

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The same solve command can be used to solve equations which have several unknowns. However, if there are multiple unknowns, you need to specify which one you want Maple to solve for. You do this by including an additional parameter in the command. The additional term, separated by a comma, is the variable you are solving for.

> solve( 3*x + 4*y = 17, x);

> A = (a + b + c + d)/4; solve(%,c);

In case you forget the quadratic formula, just solve for it.

> solve( a*x^2 + b*x + c = 0, x);

B1. Express each line equation into slope-intercept form (y= mx+b) using the solve command.
A. 6x 8y = 24 B. y - 3 = 2(x + 5) C. x/6 + y/18 + 1/9 = 0

B2. Given the formula F = (9/5)C + 32, find a formula for C in terms of F.

B3. Solve each equation or formula for the indicated variable.
A. 1/R = 1/R1 + 1/R2 + 1/R3 for R
B. 1/R = 1/R1 + 1/R2 + 1/R3 for R2

B4. A formula can represent a function. Its inverse is found by solving for x.
A. y = 3x - 7 B. y = (x-2)/(x+3) C .y = (x-2)3 + 7

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C. Decimal Solutions

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We saw how Maple can be used to solve equations using the solve command. However, there are many cases where it is impossible to find an exact solution. Using the fsolve command we can get a numeric answer if one exists. Maple is able to do this by using very sophisticated methods of approximating the answer.

Recall that we could not solve this polynomial when we saw it above.

> solve( x^5 - x + 1 = 0);

However, using the fsolve command, we can now find an answer in decimal form.

> fsolve( x^5 - x +1 = 0);

Sometimes there are multiple answers

> fsolve( x^4 = x^3 + 1, x);

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D. Solving The Long Way

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Although Maple can solve equations in a single step, you can also go through the same steps that you would do by hand. This is good for practicing the steps you need to take in solving a problem without going through all of the computations. Its also good for checking your homework!!

> equat := 4*(2*x + 1) - 11 = 2*x (x+1) - 3;

This adds 7 to both sides of the equation

> equat := % + ( 7=7);

This subtracts 2x from both sides

> equat := % - (2*x = 2*x);

This divides both sides by 6

> equat := %/6;

D1.. Try this problem by hand and verify the steps that Maple takes. Express in English sentences what step is performed on each of the following lines (for example Add 39 to both sides.)

> equat := x/12 + 3 = x/9 - 3;
> equat := % * lcm(12,9);
> equat := % - ( 3*x = 3*x );
> equat := % + ( 108 = 108);
> solve( equat, x );

D2. Find or create a new problem and solve it step-by-step using Maple in this same way.

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