For each line described, write an equation in(a)slope-intercept form, if possible, and(b)standard form. through (3, -5), parallel to y=4

Verified step by step guidance

1. Identify the slope of the given line. The equation \( y = 4 \) is a horizontal line, which means its slope is 0.
2. Since parallel lines have the same slope, the line we are looking for will also have a slope of 0.
3. Use the point-slope form of a line equation, \( y - y_1 = m(x - x_1) \), where \( m \) is the slope and \((x_1, y_1)\) is the point \((3, -5)\). Substitute \( m = 0 \), \( x_1 = 3 \), and \( y_1 = -5 \) into the equation.
4. Simplify the equation to get it into slope-intercept form, \( y = mx + b \).
5. Convert the slope-intercept form equation into standard form, \( Ax + By = C \), where \( A \), \( B \), and \( C \) are integers.

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is expressed as y = mx + b, where m represents the slope and b is the y-intercept. This form is useful for quickly identifying the slope of the line and where it crosses the y-axis. To write an equation in this form, one must know the slope and a point on the line.

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 solving systems of equations and analyzing the relationship between variables. Converting from slope-intercept to standard form often involves rearranging the equation.

Parallel Lines

Parallel lines have the same slope but different y-intercepts, meaning they will never intersect. When writing the equation of a line parallel to a given line, one must use the same slope as the original line. In this case, since the line y = 4 is horizontal, its slope is 0, which will be used to find the equation of the new line through the point (3, -5).