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

Verified step by step guidance

First, find the slope $ m $ of the line passing through the points $(-2, 4)$ and $(1, 3)$ using the slope formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{3 - 4}{1 - (-2)} \].

Simplify the slope expression to get the value of $ m $.

Use the point-slope form of a line equation with one of the points, for example $(-2, 4)$, and the slope $ m $: \[ y - y_1 = m(x - x_1) \].

Rewrite the equation from point-slope form into slope-intercept form $ y = mx + b $ by solving for $ y $ and simplifying.

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.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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