Problem 30 - College Algebra

Question

Use the given conditions to write an equation for each line in point-slope form and slope-intercept form. Passing through \((-2,-4)\) and \((1,-1)\)

Step-by-Step Solution

Answer

The equation of the line in point-slope form is \(y=x-2\), and in slope-intercept form, it is \(y=x-2\).

Step 1: Find the slope

Firstly, the slope of the line must be calculated. The slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) can be found using the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Here, \((x_1,y_1)=(-2,-4)\) and \((x_2,y_2)=(1,-1)\), so the slope becomes \(m = \frac{-1 - (-4)}{1 - (-2)} = 1\).

Step 2: Write the point-slope form

Now, substitute the slope and one point \((x_1, y_1)\) into the point-slope form formula. That yields \(y-y_1=m(x-x_1)\). Insert the values: \(y - (-4) = 1(x - (-2))\), which reduces to \(y + 4 = x + 2\), finally simplified to \(y = x - 2\).

Step 3: Write the slope-intercept form

Finally, substitute the slope and the y-intercept into the slope-intercept form formula. In this case, the y-intercept is the point where the line crosses the y-axis. By looking at the equation from Step 2, the y-intercept \(b\) is -2, and the slope \(m\) is 1. With those values, the slope-intercept form of the line is \(y=1x-2\), which can be further simplified to \(y=x-2\).

Key Concepts

Point-Slope Form

Point-slope form is a convenient way of writing the equation for a line when you have a point on the line and its slope. The general formula for point-slope form is given by \(y - y_1 = m(x - x_1)\), where \((x_1, y_1)\) is the point on the line, and \(m\) represents the slope.

For instance, with a slope of 1 and a point (-2, -4), you substitute these values into the formula, ending up with \(y - (-4) = 1(x - (-2))\). Algebraic simplification gives you the equation \(y = x - 2\), as seen in the textbook solution. It's a straightforward way to draft a line's equation, and understanding this concept enables students to easily model linear relationships from real-world scenarios.

Slope-Intercept Form

The slope-intercept form is another popular way of expressing the equation of a line and is particularly useful when you're looking to immediately identify the slope and y-intercept of the line. It is written as \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept, which is the point where the line crosses the y-axis.

In our example, taking the slope, 1, from the calculation and the y-intercept, -2, from the simplified point-slope equation, you place these into the slope-intercept formula to get \(y = 1x - 2\), which simplifies to the very concise equation \(y = x - 2\). This is the ideal form for quickly graphing the line or determining how the line will behave without further calculations.

Slope Calculation

The slope is a measure of how steep a line is and the direction it tilts. The calculation of the slope is a critical step in defining the equation of a line and is determined using the coordinates of two distinct points on the line. The formula to find the slope \(m\) is \(m = \frac{y_2 - y_1}{x_2 - x_1}\).

To calculate the slope for our exercise, use the points (-2, -4) and (1, -1). By plugging these into the formula, we find that \(m = \frac{-1 - (-4)}{1 - (-2)} = \frac{3}{3} = 1\). Slope calculation forms the basis for both point-slope and slope-intercept forms and is essential for understanding the dynamics of linear relationships.