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.

Solution Steps

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 \(a \geq 0\).

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 \(-\frac{5}{4}\).

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

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 \(m = -\frac{5}{4}\).

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

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 \(y\):

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

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:

\[ 5x + 4y = 0 \]

The coefficients \(5\) and \(4\) have no common factors, and \(a = 5 \geq 0\).

Final Answer