Smallest number for given remainders
Problem
Find the smallest number which when divided by 2 the remainder is 1, when divided by 3 the remainder is 2, when divided by 4 the remainder is 3, when divided by 5 the remainder is 4, and when divided by 6 the remainder is 5.
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Arithmetic, geometric, and harmonic progressions
Elements
a1 = value of the first term
am = value of any term after the first term but before the last term
an = value of the last term
n = total number of terms
m = mth term after the first but before nth
d = common difference of arithmetic progression
r = common ratio of geometric progression
S = sum of the 1st n terms
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Example 04 - Simultaneous Non-Linear Equations of Three Unknowns
Problem
Solve for x, y, and z from the following system of equations.
$x(y + z) = 12$ → Equation (1)
$y(x + z) = 6$ → Equation (2)
$z(x + y) = 10$ → Equation (3)
Example 03 - Simultaneous Non-Linear Equations of Three Unknowns
Problem
Find the value of x, y, and z from the given system of equations.
$x(x + y + z) = -36$ → Equation (1)
$y(x + y + z) = 27$ → Equation (2)
$z(x + y + z) = 90$ → Equation (3)
Example 02 - Simultaneous Non-Linear Equations of Three Unknowns
Problem
Find the value of x, y, and z from the following equations.
$xy = -3$ → Equation (1)
$yz = 12$ → Equation (2)
$xz = -4$ → Equation (3)
System of Equations
System of Linear Equations
The number of equations should be at least the number of unknowns in order to solve the variables. System of linear equations can be solved by several methods, the most common are the following,
1. Method of substitution
2. Elimination method
3. Cramer's rule
Many of the scientific calculators allowed in board examinations and class room exams are capable of solving system of linear equations of up to three unknowns.
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Example 01 - Simultaneous Non-Linear Equations of Three Unknowns
Problem
Solve for x, y, and z from the following simultaneous equations.
$z^x \, z^y = 100\,000$ ← equation (1)
$(z^x)^y = 100\,000$ ← equation (2)
$\dfrac{z^x}{z^y} = 10$ ← equation (3)
Sum and product of roots of quadratic equation
Problem
Find the sum and product of roots of the quadratic equation x2 - 2x + 5 = 0.
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Quadratic equation whose roots are reciprocals of the roots of another quadratic equation
Problem
Determine the equation whose roots are the reciprocals of the roots of the equation 3x2 - 13x - 10 = 0.
The student made a mistake in copying the coefficient of x and the constant term
Problem
In a quadratic equation problem, one student made a mistake in copying the coefficient of x and got roots of 3 and -2. Another student made a mistake in copying the constant term and got the roots of 3 and 2. What are the correct roots?