You already know that a system of two linear equations can be solved by substitution, elimination by equating, or elimination by addition and subtraction. Now it is time to put that to the test with more complicated systems: brackets, denominators, three unknowns... and even systems that are not linear.
Before applying a method, simplify each equation separately: expand the brackets and multiply by the lowest common multiple of the denominators to get rid of the fractions.
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Expand the brackets and solve the following linear system: $\begin{cases} 2(x + 1) - 3(y - 2) = 16 \\ 5x + y = 3 \end{cases}$
Expand the brackets and solve the following linear system:
With three equations and three unknowns, you use elimination to remove unknowns one by one until one equation is left with a single unknown.
When the system does not come already in echelon form, the first step is to put it into that form: choose an unknown and, using elimination, remove it from the second and third equations in turn, until the third equation is left with a single unknown.
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Solve the following triangular system of three equations with three unknowns: $\begin{cases} 2x - y + 3z = 14 \\ 5y - 2z = -3 \\ 4z = 16 \end{cases}$
Solve the following triangular system of three equations with three unknowns:
If one of the equations is not linear (it has $x^2$, $x \cdot y$...), you normally isolate one unknown in the linear equation and substitute it into the other. The system becomes a single quadratic equation.
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Solve the following non-linear system, formed by the equation of a parabola and a line: $\begin{cases} y = x^2 - 4x + 3 \\ y = x - 1 \end{cases}$
Solve the following non-linear system, formed by the equation of a parabola and a line:
$\begin{cases} y = x^2 - 4x + 3 \\ y = x - 1 \end{cases}$
Find the points of intersection between the hyperbola and the line given by the following non-linear system: $\begin{cases} x \cdot y = 12 \\ 2x + y = 10 \end{cases}$
Find the points of intersection between the hyperbola and the line given by the following non-linear system:
$\begin{cases} x \cdot y = 12 \\ 2x + y = 10 \end{cases}$
Extension: Changes of variable: exponential, logarithmic and irrational systems
Some systems become much simpler with a change of variable: you replace a complicated expression with a new letter, solve the linear system that results and, at the end, undo the change.
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Solve the following system of equations with algebraic fractions by applying a suitable change of variable:…
Solve the following system of equations with algebraic fractions by applying a suitable change of variable: