In 4th ESO you solved systems of two equations with two unknowns by substitution, equating or elimination. In Bachillerato those same three methods are still the foundation, but the systems get more complicated: exponentials, logarithms, roots and fractions appear, and sometimes there are three unknowns instead of two. The key idea never changes: reduce the system to something you already know how to solve.
Choosing the method according to the form of the system
There is no single "best" method: there is one that is more convenient for each system. If one unknown is almost already isolated, use substitution. If the two equations look very similar, use equating or elimination.
Practice★☆☆Not started
Find the value of the unknowns $x$, $y$, $z$ in the following system of linear equations, which is already in echelon form: $2x - y + 3z = 14$ $3y - z = 5$ $2z = 8$
Find the value of the unknowns $x$, $y$, $z$ in the following system of linear equations, which is already in echelon form:
Solve the following exponential system of equations by expressing both sides of each equation as powers with the same base: $2^{x+y} = 32$ $3^{x-y} = 9$
Solve the following exponential system of equations by expressing both sides of each equation as powers with the same base:
Apply the definition of a logarithm to set up an equivalent linear system and solve the following system of equations: $\log_2(x + y) = 3$ $x - 2y = 2$
Apply the definition of a logarithm to set up an equivalent linear system and solve the following system of equations:
In a system with algebraic fractions it is useful to call $u=\dfrac{1}{x}$, $v=\dfrac{1}{y}$ if each unknown appears only in denominators: the system then becomes linear again in $u$, $v$.
Practice★☆☆Not started
Find the solution to the following system of equations involving radical expressions: $\sqrt{x} + \sqrt{y} = 5$ $x - y = 5$
Find the solution to the following system of equations involving radical expressions:
Solve the following system of equations containing algebraic fractions, clearly stating the initial restrictions on the unknowns: $\frac{2}{x} + \frac{3}{y} = 2$…
Solve the following system of equations containing algebraic fractions, clearly stating the initial restrictions on the unknowns:
With three unknowns and three equations, substitution and equating become slow and it is easy to make mistakes. Elimination, applied in an orderly way, is the basis of Gaussian elimination: eliminating unknowns one by one until the system is left in "echelon form", like the one in the warning above.
Practice★☆☆Not started
Apply Gaussian elimination to transform and solve the following system of three linear equations in three unknowns: $x + y + z = 6$ $x - y + 2z = 5$ $x + y - z = 0$
Apply Gaussian elimination to transform and solve the following system of three linear equations in three unknowns:
Solve the following system of linear equations using Gaussian elimination and state whether it is a consistent independent system: $2x - y + 3z = 1$ $x + 2y - z = 4$…
Solve the following system of linear equations using Gaussian elimination and state whether it is a consistent independent system: