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
7

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.
kk
kk
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}
kk
kk
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.
kk
kk
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
kk
kk
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
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.
kk
kk
kChanging Coefficients
You can change a coefficient either before or after you register it by pressing w.
uu
uu
uTo change a coefficient before registering it with w
Press the Akey to clear the current value and then input another one.
uu
uu
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.
kk
kk
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
GG
G0):
• Quadratic: ax2+ bx + c= 0
• Cubic: ax3+ bx2+ cx + d= 0
kk
kk
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
kk
kk
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.
kk
kk
kMultiple root (1 or 2) solutions or imaginary number
solutions
The following examples illustrate how multiple-root solutions and imaginary
number solutions are handled.
uu
uu
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)
uu
uu
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
kk
kk
kChanging Coefficients
You can change a coefficient either before or after you register it by pressing w.
uu
uu
uTo change a coefficient before registering it with w
Press the Akey to clear the current value and then input another one.
uu
uu
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.
kk
kk
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.
kk
kk
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.
uu
uu
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
w
P.408
1
H = VT – –– GT2
2

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.
uu
uu
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

109
• Solve uses Newton’s method to calculate approximations. The following can
sometimes occur when this method is used.
—Solutions may be impossible to obtain for certain initial estimated values.
Should this happen, try inputting another value that you assume to be in
the vicinity of the solution and perform the calculation again.
—The calculator may be unable to obtain a solution, even though a solution
exists.
• Due to certain idiosyncrasies of Newton’s method, solutions for the following
types of functions tend to be difficult to calculate.
—Periodic functions (i.e. y= sinx– a)
—Functions whose graph produce sharp slopes (i.e. y= ex, y= 1/ x)
—Inverse proportion expressions and other discontinuous functions.
• Note that the Solve function can also be used in the ALGBR Mode to display
results using natural display notation.
Solve Calculations 7 - 4
P.355

110
7-5 What to Do When an Error Occurs
uu
uu
uError during coefficient value input
Press the Akey to clear the error and return to the value that was registered for
the coefficient before you input the value that generated the error. Try inputting a
new value again.
uu
uu
uError during calculation
Press the Akey to clear the error and display coefficient a. Try inputting values
for the coefficients again.