Linear Equation 0Beb88

1. **Problem Statement:** We need to find the linear equation that matches a graph described as a straight line with a positive slope, crossing the y-axis at about $-3$, and rising gently to the right. 2. **Recall the slope-intercept form of a linear equation:** $$y = mx + b$$ where $m$ is the slope and $b$ is the y-intercept. 3. **Analyze the graph description:** - The line crosses the y-axis at about $-3$, so $b \approx -3$. - The slope is positive and gentle, meaning $m$ is positive and less than 1. 4. **Check each option:** - a. $y = \frac{1}{2}x - 3$: slope $m=\frac{1}{2}$ (positive, gentle), y-intercept $b=-3$. - b. $y = -4x - 2$: slope $m=-4$ (negative, steep), y-intercept $b=-2$. - c. $y = 4x + 3$: slope $m=4$ (positive, steep), y-intercept $b=3$. - d. $y = \frac{1}{4}x - 4$: slope $m=\frac{1}{4}$ (positive, gentle), y-intercept $b=-4$. 5. **Compare y-intercepts:** The graph crosses at about $-3$, so $b=-3$ fits best. 6. **Compare slopes:** Both $\frac{1}{2}$ and $\frac{1}{4}$ are positive and gentle slopes, but the graph is described as rising gently, and $\frac{1}{2}$ is a bit steeper than $\frac{1}{4}$. 7. **Conclusion:** The best match is option a: $$y = \frac{1}{2}x - 3$$ **Final answer:** $y = \frac{1}{2}x - 3$