1. Problem: Find the x-intercept and y-intercept of the equation $4y = 2x - 1$. - To find the x-intercept, set $y=0$ and solve for $x$. - To find the y-intercept, set $x=0$ and solve for $y$. Step 1: Find x-intercept Set $y=0$: $$4(0) = 2x - 1 \implies 0 = 2x - 1 \implies 2x = 1 \implies x = \frac{1}{2}$$ So, x-intercept is $\left(\frac{1}{2}, 0\right)$. Step 2: Find y-intercept Set $x=0$: $$4y = 2(0) - 1 \implies 4y = -1 \implies y = -\frac{1}{4}$$ So, y-intercept is $(0, -\frac{1}{4})$. --- 2. Problem: Find the x-intercept and y-intercept of the equation $4x - 3 = 2y$. Rewrite as $2y = 4x - 3$. Step 1: Find x-intercept ($y=0$): $$0 = 4x - 3 \implies 4x = 3 \implies x = \frac{3}{4}$$ So, x-intercept is $\left(\frac{3}{4}, 0\right)$. Step 2: Find y-intercept ($x=0$): $$2y = 4(0) - 3 = -3 \implies y = -\frac{3}{2}$$ So, y-intercept is $(0, -\frac{3}{2})$. --- 3. Problem: Find the x-intercept and y-intercept of the equation $2x - \frac{2}{3} = \frac{3}{4} y + 3$. Step 1: Rearrange to isolate $y$ terms: $$\frac{3}{4} y = 2x - \frac{2}{3} - 3 = 2x - \frac{2}{3} - \frac{9}{3} = 2x - \frac{11}{3}$$ Step 2: Find x-intercept ($y=0$): Set $y=0$: $$0 = 2x - \frac{11}{3} \implies 2x = \frac{11}{3} \implies x = \frac{11}{6}$$ So, x-intercept is $\left(\frac{11}{6}, 0\right)$. Step 3: Find y-intercept ($x=0$): $$\frac{3}{4} y = 2(0) - \frac{11}{3} = -\frac{11}{3} \implies y = -\frac{11}{3} \times \frac{4}{3} = -\frac{44}{9}$$ So, y-intercept is $(0, -\frac{44}{9})$. --- 4. Problem: Solve for $y$ in terms of $x$ for $3x - 2y = 6$. Step 1: Isolate $y$: $$-2y = 6 - 3x \implies y = \frac{3x - 6}{2}$$ Step 2: Simplify: $$y = \frac{3}{2} x - 3$$ --- 5. Problem: Solve for $y$ in terms of $x$ for $x - 2y = 7$. Step 1: Isolate $y$: $$-2y = 7 - x \implies y = \frac{x - 7}{2}$$ --- 6. Problem: Solve for $y$ in terms of $x$ for $5x + 2y = 0$. Step 1: Isolate $y$: $$2y = -5x \implies y = -\frac{5}{2} x$$ --- 7. Problem: Find the distance between points $(2, -5)$ and $(7, 4)$. Step 1: Use distance formula: $$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$ Step 2: Substitute values: $$d = \sqrt{(7 - 2)^2 + (4 - (-5))^2} = \sqrt{5^2 + 9^2} = \sqrt{25 + 81} = \sqrt{106}$$ Step 3: Simplify if possible. $106$ factors as $2 \times 53$, no perfect squares, so distance is $\sqrt{106}$. --- Final answers: 6. x-intercept: $\left(\frac{1}{2}, 0\right)$, y-intercept: $(0, -\frac{1}{4})$ 8. x-intercept: $\left(\frac{3}{4}, 0\right)$, y-intercept: $(0, -\frac{3}{2})$ 10. x-intercept: $\left(\frac{11}{6}, 0\right)$, y-intercept: $(0, -\frac{44}{9})$ 12. $y = \frac{3}{2} x - 3$ 14. $y = \frac{x - 7}{2}$ 16. $y = -\frac{5}{2} x$ 18. Distance = $\sqrt{106}$