Problem 33 - College Algebra

Question

Use the given conditions to write an equation for each line in point-slope form and slope-intercept form. Passing through \((-3,-1)\) and \((4,-1)\)

Step-by-Step Solution

Answer

The slope of the line between points \((-3,-1)\) and \((4,-1)\) equals 0 suggesting a horizontal line. The point-slope form of the line equation is \(y + 1 = 0(x + 3)\) simplifying to \(y = -1\). The slope-intercept form is also \(y = -1\), 'c' equals -1.

Step 1: Calculating the Slope of the Line

Calculate the slope of the line using the given points \((-3,-1)\) and \((4,-1)\) using the formula \(m=(y_2-y_1)/(x_2-x_1)\).

Step 2: Writing the Point-Slope Form Equation

Now use the slope and one of the points, say \((-3,-1)\) to write the point-slope form of the equation as \(y - y_1 = m(x - x_1)\).

Step 3: Converting to the Slope-Intercept Form Equation

Transform the point-slope form into the slope-intercept form using algebraic manipulation, achieving \(y = mx + c\). You can consider that for a horizontal line \(c = y_1\).

Step 4: Finding the y-intercept c

Use the slope-intercept form and one of the points, again let’s use \((-3,-1)\), to solve for c.

Key Concepts

Point-Slope Form

The point-slope form is a popular way to write the equation of a line when you know a specific point on the line and the slope. It's expressed as \[ y - y_1 = m(x - x_1) \] where \( (x_1, y_1) \) is a point on the line, and \( m \) is the slope. This form is very handy because it clearly shows how the line passes through the given point.

• Choose any point, like \((-3, -1)\) from the original problem.
• Once you calculate the slope, insert it into the equation along with the chosen point.

By using this form, you get a clear insight into how variations in \( x \) affect \( y \). For our example, after calculating the slope, substitute to find the line's equation in point-slope form.

Slope-Intercept Form

The slope-intercept form of a line is one of the most common ways to represent a linear equation. It's easy to understand as it highlights both the slope and the y-intercept directly in its form: \[ y = mx + c \] where \( m \) stands for slope and \( c \) stands for the y-intercept. This form allows for quick graphing since once \( m \) and \( c \) are known, you can see where the line crosses the y-axis and its steepness.

• After finding the slope, manipulate the point-slope form to this slope-intercept form.
• Plug in any known point to solve for \( c \), making it possible to completely frame the equation.

For the points \((-3, -1)\) and \((4, -1)\), following these steps allows you to clearly express the equation of the line in this form.

Slope Calculation

Calculating the slope is an essential first step when writing the equation of a line. Slope, denoted by \( m \), measures how steep the line is. The formula used to calculate slope given two points \((x_1, y_1)\) and \((x_2, y_2)\) is: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

• This formula calculates the change in \( y \) (rise) over the change in \( x \) (run).
• For the given points in the problem, since both points \((-3, -1)\) and \((4, -1)\) have the same \( y \)-value, you will find that \( m = 0 \).

Recognizing that the slope is zero here indicates that the line is horizontal, which greatly simplifies further calculations.

Horizontal Line Equation

A horizontal line is special because its slope is always zero. This directly influences its equation. For any horizontal line, the equation takes a simple form: \[ y = c \] where \( c \) is the constant y-value across the entire line.

• This is because no matter what \( x \) value you choose, \( y \) will always remain the same.
• From the given points, since \( y \) is \(-1\) for both \(-3\) and \(4\), the line is \( y = -1 \).

Horizontal lines are straightforward and quickly recognized on a graph, involving minimal calculations once identified. Such understanding helps in simplifying complex problems efficiently.