Systems of linear equations, graphing and substitution
What a system of two linear equations is, what its solution looks like on a graph, and how to solve it by substitution.
One equation isn't enough
The equation \(x + y = 10\) has infinitely many solutions: \((1, 9)\),\((4, 6)\),\((2.5, 7.5)\), and so on. With two unknowns, one piece of information can't pin them down.
Add a second equation, say \(x - y = 2\), and only one pair satisfies both: \((6, 4)\).
The picture: two lines
The graph of a linear equation in \(x\) and \(y\) is a line: every point on it is a solution of that equation. A solution of the system has to be on both lines, so it's the point where they intersect.
Graphing shows what's going on, but reading an exact intersection off a sketch is hard when it isn't at whole numbers. That's why we need algebraic methods.
Two lines can meet in one of three ways:
they intersect once: exactly one solution;
they are parallel: no solution;
they are the same line: infinitely many solutions.
The substitution method
Practice
Quick check
Which ordered pair solves \(\begin{cases} x + y = 5 \\ x - y = 1 \end{cases}\)?
\(3 + 2 = 5\) and \(3 - 2 = 1\). The other pairs satisfy at most one equation.
Quick check
On a graph, what does the solution of a system of two linear equations correspond to?
The solution has to lie on both lines, so it's their point of intersection.
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