11. Algebra II: Coordinate Plane, Slope, Lines, and Systems

The coordinate plane turns algebra into pictures. A point \((x,y)\) tells you how far to move horizontally and vertically from the origin.

Slope is steepness:
\[ m=\frac{y_2-y_1}{x_2-x_1}. \]
From \((2,3)\) to \((6,11)\), slope is \((11-3)/(6-2)=8/4=2\).

The most common line form is
\[ y=mx+b, \]
where \(m\) is slope and \(b\) is the y-intercept. In \(y=-3x+4\), the slope is \(-3\) and the y-intercept is 4.

A system of equations asks for a point that makes both equations true. Solve:
\[ \begin{cases} y=x+2\\ x+y=10 \end{cases} \]
Substitute \(y=x+2\): \(x+x+2=10\), so \(2x=8\), \(x=4\), and \(y=6\). The solution is \((4,6)\).

Common mistake: giving only \(x\) for a system. A complete system answer usually gives both coordinates.