Question

1. the equation $-6x + 10y = 210$ models the linear relationship. place a check in the box of a key feature that can be determined using the equation.

domain	range	$x$-intercept	$y$-intercept	slope	increasing /decreasing

1. write the equation $-6x + 10y = 210$ in slope-intercept form.
2. what domain makes sense for the context?
3. what range makes sense for the context?
4. determine the $x$-intercept.
5. interpret the $x$-intercept.
6. determine the slope.
7. interpret the slope.

Explanation:

Step1: Rewrite to slope-intercept form

Start with $-6x + 10y = 210$. Isolate $10y$:
$10y = 6x + 210$
Divide by 10:
$y = \frac{3}{5}x + 21$

Step2: Identify calculable features (Q3)

From the equation, we can find x-intercept, y-intercept, slope, and increasing/decreasing behavior without context. Domain and range need context.

Step3: Calculate x-intercept (Q7)

Set $y=0$ in original equation:
$-6x + 10(0) = 210$
$-6x = 210$
$x = -35$

Step4: Calculate slope (Q9)

From slope-intercept form, slope is the coefficient of $x$.

Step5: Address context-dependent questions (Q5, Q6)

No context provided, so use non-negative values as a common practical context (e.g., quantities can't be negative).

Step6: Interpret intercept and slope (Q8, Q10)

Relate intercept to when the other variable is 0; slope to rate of change.

Answer:

1. Checked features:

• x-intercept
• y-intercept
• Slope
• Increasing/Decreasing

1. $y = \frac{3}{5}x + 21$
2. Assuming a practical context (e.g., non-negative quantities), the domain is $x \geq -35$; if only non-negative x is valid, $x \geq 0$.
3. Assuming a practical context (e.g., non-negative quantities), the range is $y \geq 21$; if paired with $x \geq 0$, $y \geq 21$.
4. $(-35, 0)$
5. The x-intercept $(-35, 0)$ means when $y=0$, $x=-35$; in a practical context, this could represent a baseline value where the y-quantity is zero.
6. $\frac{3}{5}$
7. The slope $\frac{3}{5}$ means for every 1-unit increase in $x$, $y$ increases by $\frac{3}{5}$ units; the relationship is increasing.