To solve these equations and sketch the graphs using intercepts, we'll calculate both the x- and y-intercepts for each equation. To do this:

• x-intercept: Set $y = 0$ and solve for $x$.
• y-intercept: Set $x = 0$ and solve for $y$.

1) $x + 5y = 25$

• x-intercept: $x + 5(0) = 25 \implies x = 25$ The x-intercept is $(25, 0)$.

• y-intercept: $0 + 5y = 25 \implies y = 5$ The y-intercept is $(0, 5)$.

2) $3x - 4y = 16$

• x-intercept: $3x - 4(0) = 16 \implies 3x = 16 \implies x = \frac{16}{3} \approx 5.33$ The x-intercept is $\left( \frac{16}{3}, 0 \right)$.

• y-intercept: $3(0) - 4y = 16 \implies -4y = 16 \implies y = -4$ The y-intercept is $(0, -4)$.

3) $3x + y = -3$

• x-intercept: $3x + 0 = -3 \implies x = -1$ The x-intercept is $(-1, 0)$.

• y-intercept: $3(0) + y = -3 \implies y = -3$ The y-intercept is $(0, -3)$.

4) $2x + y = -5$

• x-intercept: $2x + 0 = -5 \implies x = -\frac{5}{2} = -2.5$ The x-intercept is $\left( -\frac{5}{2}, 0 \right)$.

• y-intercept: $2(0) + y = -5 \implies y = -5$ The y-intercept is $(0, -5)$.

5) $9x - 4y = -20$

• x-intercept: $9x - 4(0) = -20 \implies 9x = -20 \implies x = -\frac{20}{9} \approx -2.22$ The x-intercept is $\left( -\frac{20}{9}, 0 \right)$.

• y-intercept: $9(0) - 4y = -20 \implies -4y = -20 \implies y = 5$ The y-intercept is $(0, 5)$.

Summary of Intercepts:

1. $x + 5y = 25$ → x-intercept: $(25, 0)$, y-intercept: $(0, 5)$
2. $3x - 4y = 16$ → x-intercept: $\left( \frac{16}{3}, 0 \right)$, y-intercept: $(0, -4)$
3. $3x + y = -3$ → x-intercept: $(-1, 0)$, y-intercept: $(0, -3)$
4. $2x + y = -5$ → x-intercept: $\left( -\frac{5}{2}, 0 \right)$, y-intercept: $(0, -5)$
5. $9x - 4y = -20$ → x-intercept: $\left( -\frac{20}{9}, 0 \right)$, y-intercept: $(0, 5)$

Would you like any specific details on graphing these, or further clarifications?

Relative questions:

1. How do you calculate the slope of each line from the intercepts?
2. Can you find the equation of a line perpendicular to one of these?
3. What happens if you rewrite these equations in slope-intercept form?
4. How do intercepts help in understanding the behavior of linear equations?
5. How would you find the intersection points of these lines?

Tip:

When using intercepts, plotting the points on a graph and drawing a line between them helps visualize how the equation behaves across different values of $x$ and $y$.