For each line described, write an equation in
(a)slope-intercept form, if possible, and
(b)standard form.
through $(-2,4)$ and $(1,3)$

Verified step by step guidance

1. First, find the slope \( m \) of the line passing through the points \((-2, 4)\) and \((1, 3)\) using the slope formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{3 - 4}{1 - (-2)} \].
2. Simplify the slope expression to get the value of \( m \).
3. Use the point-slope form of a line equation with one of the points, for example \((-2, 4)\), and the slope \( m \): \[ y - y_1 = m(x - x_1) \].
4. Rewrite the equation from point-slope form into slope-intercept form \( y = mx + b \) by solving for \( y \) and simplifying.
5. Convert the slope-intercept form into standard form \( Ax + By = C \) by rearranging terms so that \( x \) and \( y \) are on one side and the constant on the other, ensuring \( A, B, C \) are integers and \( A \) is non-negative.

Key Concepts

Slope of a Line

The slope measures the steepness of a line and is calculated as the change in y-values divided by the change in x-values between two points. For points (-2, 4) and (1, 3), slope = (3 - 4) / (1 - (-2)) = -1/3. Understanding slope is essential for writing the equation of a line.

Slope-Intercept Form

The slope-intercept form of a line is y = mx + b, where m is the slope and b is the y-intercept. After finding the slope, substitute one point to solve for b. This form clearly shows the slope and where the line crosses the y-axis.

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 or graphing. Converting from slope-intercept form involves rearranging terms and clearing fractions.