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\(. To do this, solve for \)y\(: subtract \)8x\( from both sides to get \)5y = -8x + 3\(, then divide both sides by 5 to get \(y = -\frac{8}{5}x + \frac{3}{5}\). The slope \)m$ of the given line is \(-\frac{8}{5}\).
2. Find the slope of the line perpendicular to the given line. Recall that perpendicular slopes are negative reciprocals of each other. So, if the original slope is \(-\frac{8}{5}\), the perpendicular slope \(m_{\perp}\) is \(\frac{5}{8}\).
3. Use the point-slope form of a line equation with the point \((0, 5)\) and the perpendicular slope \(\frac{5}{8}\). The point-slope form is \(y - y_1 = m(x - x_1)\), so substitute to get \(y - 5 = \frac{5}{8}(x - 0)\).
4. Convert the equation from point-slope form to slope-intercept form by simplifying: \(y - 5 = \frac{5}{8}x\) becomes \(y = \frac{5}{8}x + 5\). This is the slope-intercept form \(y = mx + b\).
5. Rewrite the slope-intercept form into standard form \(Ax + By = C\) by eliminating fractions and rearranging terms. Multiply both sides by 8 to clear the denominator: \$8y = 5x + 40\(. Then, rearrange to get \)-5x + 8y = 40\(. If desired, multiply both sides by \)-1\( to have a positive \)x\( coefficient: \)5x - 8y = -40$.

Key Concepts

Slope-Intercept Form of a Line

The slope-intercept form is y = mx + b, where m is the slope and b is the y-intercept. It clearly shows the rate of change and where the line crosses the y-axis, making it easy to graph and understand the line's behavior.

Perpendicular Lines and Their Slopes

Two lines are perpendicular if the product of their slopes is -1. This means the slope of a line perpendicular to another with slope m is -1/m. Understanding this helps find the slope of the required line when given another line's equation.

Standard Form of a Line

The standard form of a line is Ax + By = C, where A, B, and C are integers, and A ≥ 0. This form is useful for solving systems of equations and provides a clear, consistent way to represent linear equations.