Casio CFX-9970G User manual

Equation Calculations

Your graphic calculator can solve the following three types of

equations:

•Linear equations with two to six unknowns

• Quadratic equations

•Cubic equations

7-1 Before Beginning an Equation Calculation

7-2 Linear Equations with Two to Six Unknowns

7-3 Quadratic and Cubic Equations

7-4 Solve Calculations

7-5 What to Do When an Error Occurs

Chapter

100

7-1 Before Beginning an Equation Calculation

Before beginning an equation calculation you have to first enter the correct mode,

and you must also clear the equation memories of any data that might be left over

from a previous calculation.

kEntering an Equation Calculation Mode

In the Main Menu, select the EQUA icon to enter the Equation Mode.

• {SIML} ... {linear equation with two to six unknowns}

• {POLY} ... {quadratic or cubic equation}

• {SOLV} ... {solve calculation}

kClearing Equation Memories

1. Enter the equation calculation mode (SIML or POLY) you want to use and

perform the function key operation required for that mode.

• In the case of the SIML Mode (1), use function keys 1(2) through 5(6)

to specify the number of unknowns.

• In the case of the POLY Mode (2), use function keys 1(2) or 2(3) to

specify the degree of the polynomial.

• If your pressed 3(SOLV), advance directly to step 2.

2. Press 2(DEL).

3. Press 1(YES) to delete the applicable equation memories, or 6(NO) to

abort the operation without deleting anything.

101

7-2 Linear Equations with Two to Six Unknowns

You can use the procedures described here to solve linear equations with

unknowns that match the following formats:

Two unknowns a1x + b1y = c1

a2x + b2y = c2

Six unknowns a1x + b1y + c1z + d1t + e1u + f1v = g1

a2x + b2y + c2z + d2t + e2u + f2v = g2

a3x + b3y + c3z + d3t + e3u + f3v = g3

a4x + b4y + c4z + d4t + e4u + f4v = g4

a5x + b5y + c5z + d5t + e5u + f5v = g5

a6x + b6y + c6z + d6t + e6u + f6v = g6

• You can also solve linear equations with three, four, and five unknowns. In

each case, the format is similar to those shown above.

kSpecifying the Number of Unknowns

While in the Equation Mode, press 1(SIML) and then specify the number of

unknowns.

• {2}/{3}/{4}/{5}/{6} ... linear equation with {2}/{3}/{4}/{5}/{6} unknowns

102

7 - 2 Linear Equations with Two to Six Unknowns

kSolving Linear Equations with Three Unknowns

Example To solve the following linear equations for x, y, and z:

4x+y–2z=–1

x+6y+3z=1

–5x+4y+z=–7

1. While in the Linear Equation Mode (SIML), press 2(3), because the linear

equations being solved have three unknowns.

2. Input each coefficient.

ewbw-cw-bw

bwgwdwbw

-fwewbw-hw

Each time you press w, the input value is registered in the highlighted cell.

Each press of winputs values in the following sequence:

coefficient a1→coefficient b1→coefficient c1→coefficient d1→

coefficient an→coefficient bn→coefficient cn→coefficient dn(n= 2 to 6)

• You can input fractions and value memory contents as coefficients.

3. After inputting the coefficients, solve the equations.

1(SOLV)

Coefficient input cells

Value being input into highlighted cell

123456

Highlighted solution cell value

103

• Internal calculations are performed using a 15-digit mantissa, but results are

displayed using a 10-digit mantissa and 2-digit exponent.

• This unit performs simultaneous linear equations by placing the coefficients

inside of a matrix. Because of this, as the coefficient matrix approaches zero,

precision in the inverse matrix is reduced and so precision in the results

produced also deteriorates. For example, the solution for a linear equation with

three unknowns would be calculated as shown below.

xa1b1c1

–1 d1

y=a2b2c2d2

za3b3c3d3

• An error occurs whenever the unit is unable to solve the equations.

• Pressing 1(REPT) returns to the initial display of the Linear Equation Mode.

Depending on the coefficients that you use, it may take considerable time for

the calculation result of simultaneous linear equations to appear on the

display. Failure of a result to appear immediately does not mean that the unit

is not functioning properly.

kChanging Coefficients

You can change a coefficient either before or after you register it by pressing w.

uTo change a coefficient before registering it with w

Press the Akey to clear the current value and then input another one.

uTo change a coefficient after registering it with w

Use the cursor keys to highlight the cell that contains the coefficient that you want

to change. Next, input the value that you want to change to.

kClearing All the Coefficients

While in the Linear Equation Mode, press the 3(CLR) function key. This

operation clears all the coefficients to zero.

Linear Equations with Two to Six Unknowns 7 - 2

104

7-3 Quadratic and Cubic Equations

This calculator can also solve quadratic and cubic equations that match the

following formats (when aGG

G0):

• Quadratic: ax2+ bx + c= 0

• Cubic: ax3+ bx2+ cx + d= 0

kSpecifying the Degree of an Equation

While in the Equation Mode, press 2(POLY) and then specify the degree of the

equation.

• {2}/{3} ... {quadratic}/{cubic} equation

kSolving a Quadratic or Cubic Equation

Example To solve the following cubic equation:

x3– 2x2– x+ 2 = 0

1. Press 2(3) to enter the Cubic Equation Mode.

2. Input each coefficient.

bw-cw-bwcw

• Each time you press w, the input value is registered in the highlighted cell.

Each press of winputs values in the following sequence:

coefficient a→coefficient b→coefficient c→coefficient d

Input for coefficient dis required only for cubic equations.

• You can input fractions and value memory contents as coefficients.

3. After inputting the coefficients, press 1(SOLV) to solve the equations.

Highlighted solution cell value

105

• Internal calculations are performed using a 15-digit mantissa, but results are

displayed using a 10-digit mantissa and 2-digit exponent.

• An error occurs whenever the unit is unable to solve the equations.

• Pressing 1(REPT) returns to the initial display of the Cubic Equation Mode.

kMultiple root (1 or 2) solutions or imaginary number

solutions

The following examples illustrate how multiple-root solutions and imaginary

number solutions are handled.

uTo solve a cubic equation that produces a multiple-value solution

Example To solve the following cubic equation:

x3– 4x2+ 5x– 2 = 0

bw-ewfw-cw

1(SOLV)

uTo solve a cubic equation that produces an imaginary number

solution

Example To solve the following cubic equation:

x3+ x2+ x– 3 = 0

bwbwbw-dw

1(SOLV)

It may take considerable time for the calculation result of cubic equations to

appear on the display. Failure of a result to appear immediately does not

mean that the unit is not functioning properly.

Quadratic and Cubic Equations 7 - 3

106

kChanging Coefficients

You can change a coefficient either before or after you register it by pressing w.

uTo change a coefficient before registering it with w

Press the Akey to clear the current value and then input another one.

uTo change a coefficient after registering it with w

Use the cursor keys to highlight the cell that contains the coefficient that you want

to change. Next, input the value that you want to change to.

kClearing All the Coefficients

While in the Quadratic or Cubic Equation Mode, press the 3(CLR) function key.

This operation clears all the coefficients to zero.

7 - 3 Quadratic and Cubic Equations

107

7-4 Solve Calculations

You can determine the value of any variable you are using without going through

the trouble of solving an equation.

Input the equation, and a table of variables appears on the display. Use the table

to assign values to variables and then execute the calculation to obtain a solution

and display the value of the unknown variable.

• You cannot use the variable table in the Program Mode. When you want to use

the Solve calculation function in the Program Mode, you have to use program

commands to assign values to variables.

kEntering the Solve Calculation Mode

While in the Equation Mode, press 3(SOLV). The Solve input screen appears.

Input the expression. You can input numbers, alpha-characters, and operation

symbols. If you do not input an equals sign, the calculator assumes that the

expression is to the left of the equals sign and there is a zero to the right. To

specify a value other than zero to the right of the equals sign, you must input the

equals sign and the value.

uTo perform solve calculations

Example To calculate initial velocity of an object thrown into the air and

taking a time of 2 seconds to reach a height of 14 meters, when

gravitational acceleration is 9.8 m/S2

The following formula expresses the relationship between height H, initial velocity

V, time T, and gravitational acceleration G of a free falling object.

1. Press 2(DEL) 1(YES) to clear any previously input equations.

2. Input the equation.

aH!=aVaT-(b/c)aGaTx

P.408

H = VT – –– GT2

108

3. Input the values.

bew(H=14)

aw(V=0)

cw(T=2)

j.iw(G=9.8)

4. Press fto move the highlighting to V = 0.

5. Press 6(SOLV) to obtain the solution.

• An error occurs if you input more than one equals sign.

• “Lft” and “Rgt” indicate the left and right sides that are calculated using the

approximate value. The closer the difference between these two values is to

zero, the greater the accuracy of the result.

Solve Calculations

The solution of the function is approximated using Newton’s method.

uNewton’s method

This method is based on the assumption

that f(x)can be approximated by a linear

expression within a very narrow range.

First, a starting value (predicted value) xois

given. Using this starting value as a base,

approximate value x1is obtained, and then

the left side and right side calculation

results are compared. Next, approximate

value x1is used as the initial value to

calculate the next approximate value x2.

This procedure is repeated until the

difference between the left side and right

side calculated values is less than some

minute value.

• Solutions obtained using Newton’s method may include errors.

• To check results, plug them into the original expression and perform the

calculation.

7 - 4 Solve Calculations

Equation

Solution

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