Problem 20 - Introductory Algebra for College Students

Question

Write the point-slope form of the equation of the line satisfying each of the conditions in Exercises. Then use the point-slope form of the equation to write the slope-intercept form of the equation. Passing through \((-2,-4)\) and \((1,-1)\)

Step-by-Step Solution

Answer

The point-slope form of the equation of the line is \( y + 4 = x + 2 \), while the slope-intercept form is \( y = x - 2 \).

Step 1: Compute the Slope

The slope of the line passing through the points (-2,-4) and (1,-1) can be calculated using the formula: \( m = \frac{{y2 - y1}}{{x2 - x1}} = \frac{{-1 - (-4)}}{{1 - (-2)}} = 1.\)

Step 2: Use the point-slope form equation

Plug one of the given points into the point-slope equation to find the line equation. Using point (-2,-4), the equation becomes: \( y - (-4) = 1 * (x - (-2)) \) which simplifies to \( y + 4 = x + 2\).

Step 3: Convert to Slope-Intercept Form

The Step 2 equation \( y + 4 = x + 2 \) can be rewritten in slope-intercept form (y = mx + b) by subtracting 4 from both sides to solve for y, resulting in \( y = x - 2 \.)

Key Concepts

Slope-Intercept Form

The slope-intercept form is a quick and efficient way to represent a linear equation. It's defined as \( y = mx + b \), where \( m \) represents the slope of the line, and \( b \) is the y-intercept, the point where the line crosses the y-axis.

When you have two points, you can easily write the equation of the line in this format. To convert from point-slope form to slope-intercept form, simply isolate \( y \) on one side of the equation. For example, if you start with a point-slope form like \( y - y_1 = m(x - x_1) \), you rearrange it to the respective slope-intercept form by solving for \( y \). This is what happens in Step 3 of the solution, resulting in the final equation \( y = x - 2 \), which states that for every one unit you move right along the x-axis, you move one unit up along the y-axis.

Calculate Slope

Calculating the slope is fundamental in algebra, as it determines the direction and steepness of a line. The slope, denoted as \( m \), is computed using the change in y-coordinates over the change in x-coordinates between two points on the line. This is articulated through the slope formula \( m = \frac{{y_2 - y_1}}{{x_2 - x_1}} \).

In our exercise, the slope is found by using the coordinates of two points provided: \( (-2, -4) \) and \( (1, -1) \). Following Step 1, we apply the values to our slope formula, yielding \( m = 1 \). This calculation is crucial because the slope is the same between any two points on a straight line, which means once we have it, we can use it to form the equation of the line with any point it passes through.

Algebraic Equations

An algebraic equation is an equality involving variables and numbers. In the context of our exercise, the algebraic equation represents the equation of a line. The equation can take several forms, such as point-slope form, slope-intercept form, and standard form.

Each form has its own uses and can be converted into the others with algebraic manipulation. Point-slope form, \( y - y_1 = m(x - x_1) \), is particularly handy when you know a point on the line and its slope. To find the slope-intercept form, algebraic steps such as distributing and combining like terms are used to isolate \( y \), resulting in an equation that tells you how to draw the line if you were to graph it. The completion of Step 2 and Step 3 in the solution demonstrates this process, ultimately yielding an equation that shows how \( y \) is calculated based on \( x \).