Questions: Find an equation for the given line in the form ax+by=c, where a, b, and c are integers with no factor common to all three and a ≥ 0. Through (-8,10); parallel to 5x+4y=13 The equation of the line in the form ax+by=c, passing through (-8,10) and parallel to 5x+4y=13 is.

Find an equation for the given line in the form ax+by=c, where a, b, and c are integers with no factor common to all three and a ≥ 0. Through (-8,10); parallel to 5x+4y=13

The equation of the line in the form ax+by=c, passing through (-8,10) and parallel to 5x+4y=13 is.

Transcript text: Find an equation for the given line in the form $a x+b y=c$, where $a, b$, and $c$ are integers with no factor common to all three and $a \geq 0$. Through $(-8,10)$; parallel to $5 x+4 y=13$ The equation of the line in the form $a x+b y=c$, passing through $(-8,10)$ and parallel to $5 x+4 y=13$ is $\square$.

Solution

Solution Steps

Step 1: Understand the Problem

We need to find the equation of a line that passes through the point $(-8, 10)$ and is parallel to the line given by $5x + 4y = 13$. The equation should be in the form $ax + by = c$, where $a, b,$ and $c$ are integers with no common factors, and $a \geq 0$.

Step 2: Identify the Slope of the Given Line

The given line is $5x + 4y = 13$. To find the slope, we can rewrite this in slope-intercept form $y = mx + b$.

\[ 5x + 4y = 13 \implies 4y = -5x + 13 \implies y = -\frac{5}{4}x + \frac{13}{4} \]

The slope $m$ of the given line is $-\frac{5}{4}$.

Step 3: Use the Slope for the Parallel Line

Since parallel lines have the same slope, the line we are looking for will also have a slope of $-\frac{5}{4}$.

Step 4: Use the Point-Slope Form

We use the point-slope form of the equation of a line, which is $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is the point $(-8, 10)$ and $m = -\frac{5}{4}$.

\[ y - 10 = -\frac{5}{4}(x + 8) \]

Step 5: Simplify the Equation

First, distribute the slope on the right-hand side:

\[ y - 10 = -\frac{5}{4}x - \frac{5}{4} \cdot 8 \]

\[ y - 10 = -\frac{5}{4}x - 10 \]

Next, add 10 to both sides to isolate $y$:

\[ y = -\frac{5}{4}x \]

Step 6: Convert to Standard Form

We need to convert $y = -\frac{5}{4}x$ to the form $ax + by = c$. Multiply every term by 4 to clear the fraction:

\[ 4y = -5x \]

Rearrange to get the standard form:

\[ 5x + 4y = 0 \]