1. **Problem statement:** Find the equation of lines with given slopes and describe the meaning of $m$ and $b$ in $y=mx+b$. Also, describe and graph $y=-3x+6$ without technology. 2. **Formula:** The equation of a line in slope-intercept form is $$y = mx + b$$ where $m$ is the slope and $b$ is the y-intercept. 3. **Step 1: Equation of lines with slope $\frac{1}{2}$ (Graph 1).** - The slope $m=\frac{1}{2}$ means for every 2 units moved horizontally, the line rises 1 unit vertically. - The lines cross the y-axis at different points $b$, so general form: $$y = \frac{1}{2}x + b$$ where $b$ varies. 4. **Step 2: Equation of lines with slope $-\frac{1}{3}$ (Graph 2).** - The slope $m=-\frac{1}{3}$ means for every 3 units moved horizontally, the line falls 1 unit vertically. - The lines cross the y-axis at different points $b$, so general form: $$y = -\frac{1}{3}x + b$$ where $b$ varies. 5. **Step 3: Meaning of $m$ and $b$ in $y=mx+b$.** - $m$ is the slope: it tells how steep the line is and the direction (positive slope rises, negative slope falls). - $b$ is the y-intercept: the point where the line crosses the y-axis (when $x=0$). - To graph, start at $(0,b)$ on the y-axis, then use slope $m$ to find other points by moving right and up/down. 6. **Step 4: Describe and graph $y = -3x + 6$.** - Slope $m = -3$ means the line falls 3 units vertically for every 1 unit moved horizontally to the right. - Y-intercept $b = 6$ means the line crosses the y-axis at $(0,6)$. - Plot point $(0,6)$. - From $(0,6)$, move 1 unit right to $x=1$, then move 3 units down to $y=3$ to plot $(1,3)$. - Draw a straight line through these points. **Final answer:** - Lines with slope $\frac{1}{2}$: $$y = \frac{1}{2}x + b$$ - Lines with slope $-\frac{1}{3}$: $$y = -\frac{1}{3}x + b$$ - $m$ is slope, $b$ is y-intercept. - Graph of $y = -3x + 6$ passes through $(0,6)$ and $(1,3)$ with slope $-3$.