Write an equation (a) in standard form and (b) in slope-intercept form for each line described. through (-1, 4), parallel to x+3y=5

Verified step by step guidance

1. Identify the slope of the given line by rewriting the equation \(x + 3y = 5\) in slope-intercept form \(y = mx + b\).
2. Rewrite \(x + 3y = 5\) as \(3y = -x + 5\) and then divide by 3 to get \(y = -\frac{1}{3}x + \frac{5}{3}\).
3. Recognize that parallel lines have the same slope, so the slope of the new line is \(-\frac{1}{3}\).
4. Use the point-slope form \(y - y_1 = m(x - x_1)\) with point \((-1, 4)\) and slope \(-\frac{1}{3}\) to write the equation: \(y - 4 = -\frac{1}{3}(x + 1)\).
5. Convert the equation to standard form \(Ax + By = C\) by clearing fractions and rearranging terms.

Key Concepts

Standard Form of a Linear Equation

The standard form of a linear equation is expressed as Ax + By = C, where A, B, and C are integers, and A should be non-negative. This form is useful for easily identifying the x- and y-intercepts of the line. To convert from slope-intercept form (y = mx + b) to standard form, you can rearrange the equation to fit the Ax + By = C format.

Slope-Intercept Form

The slope-intercept form of a linear equation is given by y = mx + b, where m represents the slope of the line and b is the y-intercept. This form is particularly useful for quickly graphing the line, as it directly provides the slope and where the line crosses the y-axis. Understanding how to manipulate this form is essential for solving problems involving linear equations.

Parallel Lines

Two lines are parallel if they have the same slope but different y-intercepts. In the context of linear equations, if a line is parallel to another, it will not intersect it at any point. To find the equation of a line parallel to a given line, you can use the slope of the original line and the coordinates of a point through which the new line passes to derive its equation.