Write an equation for each line described. Give answers in standard form for Exercises 11–20 and in slope-intercept form (if possible) for Exercises 21–32. through (-2,5) having slope -4

Verified step by step guidance

1. Identify 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 a point on the line.
2. Substitute the given point \((-2, 5)\) and slope \(-4\) into the point-slope form: \( y - 5 = -4(x + 2) \).
3. Simplify the equation to get it into slope-intercept form \( y = mx + b \): Distribute the \(-4\) on the right side: \( y - 5 = -4x - 8 \).
4. Add 5 to both sides to solve for \( y \): \( y = -4x - 3 \).
5. Convert the equation to standard form \( Ax + By = C \): Add \(4x\) to both sides: \( 4x + y = -3 \).

Key Concepts

Point-Slope Form of a Line

The point-slope form of a line is expressed as y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope. This form is particularly useful for writing the equation of a line when you know a point on the line and its slope. In this case, with the point (-2, 5) and a slope of -4, you can directly substitute these values into the formula.

Standard Form of a Linear Equation

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 useful for easily identifying the x-intercept and y-intercept of the line. To convert from point-slope or slope-intercept form to standard form, you may need to rearrange the equation and eliminate fractions.

Slope-Intercept Form of a Line

The slope-intercept form of a line is written as y = mx + b, where m represents the slope and b is the y-intercept. This form is advantageous for quickly identifying the slope and where the line crosses the y-axis. If possible, converting the equation from point-slope form to slope-intercept form can provide a clearer understanding of the line's behavior.