Problem 18 - College Algebra Essentials

Question

Use the given conditions to write an equation for each line in point-slope form and slope-intercept form. Slope \(=-2,\) passing through \((0,-3)\)

Step-by-Step Solution

Answer

The line's equation in point-slope form is \(y + 3 = -2x\) and in slope-intercept form it's \(y = -2x - 3\).

Step 1: Construct the Point-Slope Form

The point-slope form of a line is given by the formula \(y - y_1 = m(x - x_1)\) where \(m\) is the slope and \((x_1, y_1)\) is a point on the line. Given the slope \(m = -2\) and the point \((0, -3)\), substitute these values into the formula to get the point-slope form equation: \(y - (-3) = -2(x - 0)\). Simplify this to get the equation \(y + 3 = -2x\).

Step 2: Rewrite the Equation into Slope-Intercept Form

The slope-intercept form of a line is given by the formula \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. To convert from point-slope form to slope-intercept form, solve the point-slope form equation for \(y\). Starting with \(y + 3 = -2x\), subtract 3 from both sides to get \(y = -2x - 3\).

Step 3: Check the Result

Now, you have the line equation in both point-slope form and slope-intercept form. The point-slope form is \(y + 3 = -2x\), and the slope-intercept form is \(y = -2x - 3\).

Key Concepts

Point-Slope Form

The point-slope form is a way to write the equation of a line given a point on the line and the slope. It's particularly useful when you want to quickly sketch or understand the behavior of a line. The formula is given by: \[ y - y_1 = m(x - x_1) \] where \(m\) is the slope and \((x_1, y_1)\) is a point on the line. This form highlights how the line moves from a specific point. For example, given a slope of -2 and a point (0, -3), the point-slope form becomes: - Substitute the slope \(m = -2\) and the point \((0, -3)\) into the formula to get: \[ y - (-3) = -2(x - 0) \] - Simplifying this will result in: \[ y + 3 = -2x \] This equation tells you that for every step to the right on the x-axis, your step on the y-axis will descend by 2 units, starting from when the y-value is increased by 3 when \(x\) is 0.

Slope-Intercept Form

The slope-intercept form is one of the most common ways to express the equation of a line. It is written as: \[ y = mx + b \] In this form, \(m\) represents the slope of the line, and \(b\) is the y-intercept, which is the point where the line crosses the y-axis. This form is handy for quickly graphing because you start at the y-intercept \(b\) and use the slope \(m\) to determine the next points of the line. To convert from point-slope form \( y + 3 = -2x \) to slope-intercept form: - Subtract 3 from both sides to isolate \(y\): \[ y = -2x - 3 \] The equation \(y = -2x - 3\) tells you that the line has a slope of -2 and crosses the y-axis at -3.

Slope of a Line

The slope of a line is a crucial concept in linear algebra. It measures the steepness and direction of the line and is typically denoted by the letter \(m\). The slope is calculated as the "rise over run," which is the change in the y-values divided by the change in the x-values. - If you have two points, \((x_1, y_1)\) and \((x_2, y_2)\), the slope \(m\) is computed by the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] - A positive slope means the line is rising from left to right, while a negative slope indicates it is falling. - A slope of zero indicates a perfectly horizontal line, whereas an undefined slope indicates a vertical line. In our example, the given slope is -2, which tells us the line falls 2 units on the y-axis for every 1 unit it moves to the right on the x-axis. Understanding the slope helps in visualizing the line behavior and is essential for constructing both the point-slope and slope-intercept forms of a linear equation.