Problem 16 - College Algebra

Question

Use the given conditions to write an equation for each line in point-slope form and slope-intercept form. Slope \(--5,\) passing through \((-4,-2)\)

Step-by-Step Solution

Answer

The equation of the line in point-slope form is \(y + 2 = -5(x + 4)\) and in slope-intercept form is \(y = -5x - 22\).

Step 1: Apply the Point-Slope Form

First, plug the given point \((-4, -2)\) and the slope \(-5\) into the point-slope formula \(y - y1 = m(x - x1)\). Thus, the equation becomes \(y - (-2) = -5(x - (-4))\).

Step 2: Simplify the Equation

Next, simplify the above equation to get the equation of the line in point-slope form. This results in \(y + 2 = -5(x + 4)\).

Step 3: Apply the Slope-Intercept Form

Now, rearrange the formula from step 2 to match the format of the slope-intercept form \(y = mx + b\). Doing so, you get \(y = -5x - 20 - 2\) which simplifies to \(y = -5x - 22\).

Key Concepts

Point-Slope Form

The Point-Slope Form is a useful way to write the equation of a line when you're given a point on the line and the slope. It is formulated as \(y - y_1 = m(x - x_1)\), where \(m\) represents the slope and \((x_1, y_1)\) is a specific point on the line. This form is particularly handy because it directly utilizes the given data: a point and a slope, making it quicker to form an equation.

• Start with identifying the slope \(m\) and the point \((x_1, y_1)\).
• Plug these values into the formula.

The result is an equation that clearly shows how much the line rises or falls between any two points. For example, with a slope of -5 and a point (-4, -2), the equation becomes \(y + 2 = -5(x + 4)\). This highlights the line's steep drop as it moves from left to right.

Slope-Intercept Form

After you have a line's equation in point-slope form, you can rearrange it into slope-intercept form, \(y = mx + b\). This form is incredibly popular because it directly shows the slope \(m\) and the y-intercept \(b\), the point where the line crosses the y-axis.

• Start with your point-slope equation.
• Distribute the slope through any parentheses.
• Solve for \(y\) to put the equation in \(y = mx + b\) form.

In our example, rearranging \(y + 2 = -5(x + 4)\) by distributing \(-5\) and then isolating \(y\), we get \(y = -5x - 22\). This beautifully shows that for every unit increase in \(x\), \(y\) decreases by 5 units, and when \(x = 0\), \(y\) is -22.

Equation of a Line

The equation of a line can be expressed in multiple forms, each useful for different scenarios. Whether you're working with the point-slope form or converting it to the slope-intercept form, you're describing the same line through numbers. This versatility helps in various applications from graphing to solving real-world problems.

• Point-slope form is great for constructing an equation with specific points and slope.
• Slope-intercept form provides a clean view of how the line behaves with respect to the axes.

In essence, both forms describe a line's behavior in relation to its slope and position. Remember that an equation like \(y = -5x - 22\) is a powerful tool allowing you to predict \(y\) for any given \(x\), making it essential for deeper mathematical understanding and practical applications.