Problem 57 - Algebra 1

Question

Write an equation of the line in slope-intercept form that passes through the two points, or passes through the point and has the given slope. $$(-2,-1),(4,2)$$

Step-by-Step Solution

Answer

The equation of the line is \(y = 0.5x + 2\).

Step 1: Calculate the slope

Slope, or \(m\), can be calculated using the formula \(m = (y_2 - y_1) / (x_2 - x_1)\). So, for points (-2,-1) and (4,2), this gives \(m = (2 - (-1)) / (4 - (-2)) = 3/6 = 0.5\)

Step 2: Use point-slope to create an equation

Use one of the points (-2, -1) in the point-slope formula \(y - y_1 = m(x - x_1)\), where \(m\) is the slope and \((x_1, y_1)\) is any point on the line. This gives us \(y - (-1) = 0.5(x - (-2))\).

Step 3: Simplify the equation into slope-intercept form

Rearrange the equation from step 2 into slope-intercept form \(y = mx + b\). This gives \(y = 0.5x + 2\).

Key Concepts

Calculating Slope

Understanding how to calculate the slope of a line is crucial in algebra and geometry. The slope is a measure of how steep a line is. To find the slope, also known as 'rise over run', between two points \( (x_1, y_1) \) and \( (x_2, y_2) \), you use the formula \( m = (y_2 - y_1) / (x_2 - x_1) \).

For example, with points \( (-2, -1) \) and \( (4, 2) \), the slope \( m \) is calculated as follows: \( m = (2 - (-1)) / (4 - (-2)) = 3 / 6 = 0.5 \).

The result is \( m = 0.5 \), indicating that for every one unit you move to the right (\