For each line described, write an equation in(a)slope-intercept form, if possible, and(b)standard form. through (0, 5), perpendicular to 8x+5y=3

Verified step by step guidance

1. Identify the slope of the given line by rewriting the equation \(8x + 5y = 3\) in slope-intercept form \(y = mx + b\).
2. The slope-intercept form is \(y = -\frac{8}{5}x + \frac{3}{5}\), so the slope \(m\) is \(-\frac{8}{5}\).
3. Find the slope of the line perpendicular to the given line. The perpendicular slope is the negative reciprocal of \(-\frac{8}{5}\), which is \(\frac{5}{8}\).
4. Use the point-slope form \(y - y_1 = m(x - x_1)\) with the point \((0, 5)\) and the perpendicular slope \(\frac{5}{8}\) to write the equation.
5. Convert the equation from point-slope form to both slope-intercept form \(y = mx + b\) and standard form \(Ax + By = C\).

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is expressed as y = mx + b, where m represents the slope of the line and b is the y-intercept. This form is useful for quickly identifying the slope and y-intercept, making it easier to graph the line. To convert an equation into this form, one typically solves for y.

Standard Form

The standard form of a linear equation is given by Ax + By = C, where A, B, and C are integers, and A should be non-negative. This form is beneficial for analyzing the relationship between x and y and is often used in systems of equations. To convert from slope-intercept form to standard form, rearranging the equation is necessary.

Perpendicular Lines

Two lines are perpendicular if the product of their slopes is -1. This means that if one line has a slope of m, the slope of the line perpendicular to it will be -1/m. Understanding this concept is crucial for finding the slope of the line that is perpendicular to a given line, which is necessary for solving the problem at hand.