06/10: Systems and Algebraic Equations Free Online FlashCards

What is a solution of an equation or system?

A solution is a value, ordered pair, or tuple that makes every original equation true.

What domain restrictions must be checked over the real numbers?

A denominator cannot be zero, a real logarithm's argument must be positive, and an even-root radicand must be nonnegative.

What does the zero-product property state?

If AB=0, then A=0 or B=0. This property turns a factored polynomial equation into simpler equations.

What is the quadratic formula?

For ax²+bx+c=0, the quadratic formula is x=(-b±√(b²-4ac))/(2a).

How does the discriminant classify quadratic solutions?

The discriminant is Δ=b²−4ac. Positive means two distinct real solutions, zero means one repeated real solution, and negative means no real solutions.

What is the standard procedure for a rational equation?

For a rational equation, state denominator restrictions, multiply by the least common denominator, solve, reject restricted or extraneous values, and check the original equation.

Why can equal exponential powers have their exponents equated?

Because b^u and b^v are equal with a valid common base b>0 and b≠1, one-to-one behavior gives u=v.

How do you solve Ab^(kx)=C when bases cannot be matched?

For Ab^(kx)=C, isolate the exponential term and take logarithms: x=ln(C/A)/(k ln b), provided the domain and sign requirements hold.

What conditions define a real logarithm?

A logarithm requires a positive argument: log_b(x) is defined for x>0, with b>0 and b≠1.

Why must logarithmic solutions be checked against the domain?

Combining logarithms can produce candidate values outside the original positive-argument domain; checking the original restrictions removes these extraneous solutions.

When is substitution a useful method for a system?

Substitution is effective when one equation already expresses a variable in terms of another; substitute that expression into the other equation.

How does elimination solve a compatible system?

Elimination adds or subtracts equations to cancel a variable. For 2x+y=7 and 3x−y=8, addition gives 5x=15 and then (x,y)=(3,1).