7. Writing Linear Equations from Evidence

Graph and table questions often ask for the equation of a line. A linear equation needs two pieces of evidence: slope and one point. The most common form is:
\[ y=mx+b. \]

If the slope is 3 and the y-intercept is \(-2\), the equation is:
\[ y=3x-2. \]

From a table, first find slope. Suppose the table includes \((0,5),(1,8),(2,11)\). Each time \(x\) increases by 1, \(y\) increases by 3, so \(m=3\). Since \(x=0\) gives \(y=5\), the y-intercept is 5. The equation is:
\[ y=3x+5. \]

From a point and a slope, substitute to find \(b\). A line has slope 2 and passes through \((4,11)\):
\[ 11=2(4)+b,\quad 11=8+b,\quad b=3. \]
So the equation is \(y=2x+3\).

Common misconception: using any visible \(y\)-value as \(b\). The y-intercept is the \(y\)-value only when \(x=0\).

Academic habit: write the evidence as "slope = ..., point = ..." before building the equation.