For each line described, write an equation in (a) slope-intercept form, if possible, and (b) standard form. through (3, -5) with slope -2.

Verified step by step guidance

1. Recall that the slope-intercept form of a line is given by the equation \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
2. Use the given slope \(m = -2\) and the point \((3, -5)\) to find the y-intercept \(b\). Substitute \(x = 3\), \(y = -5\), and \(m = -2\) into the slope-intercept form: \(-5 = -2 \times 3 + b\).
3. Solve the equation \(-5 = -6 + b\) for \(b\) by adding 6 to both sides: \(b = -5 + 6\).
4. Write the slope-intercept form of the line using the slope \(m = -2\) and the y-intercept \(b\) found in the previous step: \(y = -2x + b\).
5. To write the equation in standard form, rearrange the slope-intercept form \(y = -2x + b\) by moving all terms to one side to get \(Ax + By = C\), where \(A\), \(B\), and \(C\) are integers and \(A\) is non-negative.

Key Concepts

Slope-Intercept Form

The slope-intercept form of a line is y = mx + b, where m represents the slope and b is the y-intercept. It is useful for quickly graphing lines and understanding how the slope affects the line's steepness and direction.

Point-Slope Relationship

Given a point (x₁, y₁) and a slope m, the equation of the line can be found using y - y₁ = m(x - x₁). This form helps derive the slope-intercept form by substituting the known point and slope.

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. It is another way to express linear equations, often used for solving systems of equations or when integer coefficients are preferred.