Complete the following parts given the line 6x + 13y = −19. Enter all answers in exact, reduced form. No decimals are allowed. (a) If y decreases by 13 units, what is the corresponding change in x? (b) If x increases by 2 units, what is the corresponding change in y? (c) If x decreases by 3 units, what is the corresponding change in y? (d) If y increases by 6 units, what is the corresponding change in x?

We are given the line \(6x + 13y = -19\), and the problem asks for the corresponding changes in \(x\) or \(y\) based on certain changes in the other variable. To solve these, we can rearrange the equation to express one variable in terms of the other.

Let's solve for \(x\) in terms of \(y\):

\[
6x = -19 - 13y
\]

\[
x = \frac{-19 - 13y}{6}
\]

This equation will help us find how \(x\) changes with respect to \(y\).

Next, let's solve for \(y\) in terms of \(x\):

\[
13y = -19 - 6x
\]

\[
y = \frac{-19 - 6x}{13}
\]

Now, we can approach each part of the problem individually:

---

### (a) If \(y\) decreases by 13 units, what is the corresponding change in \(x\)?
We can determine the relationship between the changes in \(x\) and \(y\) by examining the coefficients in the original equation. The coefficient of \(x\) is 6, and the coefficient of \(y\) is 13, indicating that for a change in \(y\), the corresponding change in \(x\) is proportional to the ratio \(\frac{-13}{6}\).

If \(y\) decreases by 13 units:
\[
\Delta x = \frac{-13}{6} \times (-13) = \frac{169}{6}
\]

Thus, the corresponding change in \(x\) is \(\frac{169}{6}\).

---

### (b) If \(x\) increases by 2 units, what is the corresponding change in \(y\)?

If \(x\) increases by 2 units, we use the ratio of the coefficients \(\frac{-6}{13}\), which gives the relationship between changes in \(x\) and \(y\). So:

\[
\Delta y = \frac{-6}{13} \times 2 = \frac{-12}{13}
\]

Thus, the corresponding change in \(y\) is \(\frac{-12}{13}\).

---

### (c) If \(x\) decreases by 3 units, what is the corresponding change in \(y\)?

If \(x\) decreases by 3 units:
\[
\Delta y = \frac{-6}{13} \times (-3) = \frac{18}{13}
\]

Thus, the corresponding change in \(y\) is \(\frac{18}{13}\).

---

### (d) If \(y\) increases by 6 units, what is the corresponding change in \(x\)?

If \(y\) increases by 6 units:
\[
\Delta x = \frac{-13}{6} \times 6 = -13
\]

Thus, the corresponding change in \(x\) is \(-13\).

---

Do you want details or have any questions?

Here are 5 related questions to explore:

1. How would the corresponding changes differ if the line equation were modified?
2. Can you derive the slope of the line from the given equation?
3. How would the changes in variables differ if the coefficients of \(x\) and \(y\) were swapped?
4. What if we calculated changes for \(x\) and \(y\) using slope-intercept form?
5. How can you graphically interpret these variable changes on a coordinate plane?

**Tip:** Always express your final answer in exact, reduced form to avoid rounding errors.

Explore how to calculate the corresponding changes in x and y given the linear equation 6x + 13y = -19. This detailed breakdown covers key algebraic concepts, linear equation transformations, and ratios of coefficients. Ideal for Grades 8-10 students learning about systems of equations.

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