1. **Stating the problem:** We are given multiple linear equations and asked to analyze their slopes and intercepts, and understand how the value of $y$ changes with respect to $x$. 2. **Formula used:** The general form of a linear equation is $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept. 3. **Important rules:** - The slope $m$ indicates the rate of change of $y$ with respect to $x$. - If $m$ is positive, $y$ increases as $x$ increases. - If $m$ is negative, $y$ decreases as $x$ increases. 4. **Analyzing each equation:** - $y = \frac{2}{3}x + 5$: slope $m = \frac{2}{3}$ (positive), so $y$ increases by $\frac{2}{3}$ for each increase of 1 in $x$. - $y = x - 3$: slope $m = 1$ (positive), so $y$ increases by 1 for each increase of 1 in $x$. - $y = -\frac{3}{4}x + 3$: slope $m = -\frac{3}{4}$ (negative), so $y$ decreases by $\frac{3}{4}$ for each increase of 1 in $x$. - $y = -\frac{1}{2}x - 4$: slope $m = -\frac{1}{2}$ (negative), so $y$ decreases by $\frac{1}{2}$ for each increase of 1 in $x$. - $y = 4x + 7$: slope $m = 4$ (positive), so $y$ increases by 4 for each increase of 1 in $x$. 5. **Interpreting the given numbers:** - $\frac{4}{7}$, 7, $\frac{7}{4}$, 4 are likely slope or intercept values. 6. **Solving the equation $2y + x = 18$ for $y$:** $$ 2y + x = 18 \\ 2y = 18 - x \\ y = \frac{18 - x}{2} = 9 - \frac{x}{2} $$ Slope $m = -\frac{1}{2}$, intercept $b = 9$. 7. **Solving the equation $2y - 6x = 10$ for $y$:** $$ 2y - 6x = 10 \\ 2y = 10 + 6x \\ y = 5 + 3x $$ Slope $m = 3$, intercept $b = 5$. 8. **Matching slopes to the graph lines:** - (1) $y$ decreases by 6 means slope $m = -6$. - (2) $y$ increases by 3 means slope $m = 3$. - (3) $y$ increases by 2 means slope $m = 2$. - (4) $y$ decreases by 10 means slope $m = -10$. **Summary:** The slope determines whether $y$ increases or decreases as $x$ increases. Positive slope means increase, negative slope means decrease. The intercept $b$ is where the line crosses the y-axis. **Final answers:** - Equation $2y + x = 18$ rewritten as $y = 9 - \frac{1}{2}x$ has slope $-\frac{1}{2}$. - Equation $2y - 6x = 10$ rewritten as $y = 5 + 3x$ has slope 3. - The graph lines correspond to slopes $-6$, 3, 2, and $-10$ respectively for lines (1), (2), (3), and (4).