Problem 5 - College Algebra

Question

Use the given conditions to write an equation for each line in point-slope form and slope-intercept form. Passing through \((-8,-10)\) and parallel to the line whose equation is \(y=-4 x+3\)

Step-by-Step Solution

Answer

The equation of the line in point-slope form is \( y + 10 = -4 (x + 8) \) and in slope-intercept form is \( y = -4x - 42 \).

Step 1: Finding the slope of the given line

To find the slope of the line passing through (-8,-10) and parallel to the line whose equation is \( y=-4x+3 \), observe that the slope of the line is the coefficient of \(x\), which is -4. As parallel lines have the same slope, therefore the slope of the required line is -4.

Step 2: Applying Point-Slope Form

The formula for point-slope form is: \[ y - y_1 = m(x - x_1) \] Substituting the point (-8, -10) into \(x_{1}, y_{1}\) and -4 into \(m\), we get: \[ y - (-10) = -4 [x - (-8)] \] Simplifying we get: \[ y + 10 = -4 (x + 8) \]. This is the equation of a line in point-slope form.

Step 3: Converting to Slope-Intercept Form

Simplify our Point-Slope equation to convert it into Slope-Intercept form: \[ y + 10 = -4x - 32 \] Instead of adding 10 on both sides, subtracting 10 from both sides of the equation, we get: \[ y = -4x - 42 \]. This is the equation of a line in slope-intercept form.

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is one of the most common ways to write the equation of a line. It is expressed as \( y = mx + b \), where:

• \( y \) is the dependent variable
• \( m \) is the slope of the line
• \( x \) is the independent variable
• \( b \) is the y-intercept, or the point where the line crosses the y-axis

This form is especially useful because it clearly shows the rate of change of the line (slope) and the starting value (y-intercept). It is a straightforward way to visualize and calculate the behavior of a line. For instance, in our exercise, the slope-intercept form was determined as \( y = -4x - 42 \). Here, the slope \( m = -4 \) tells us the line decreases by 4 units for every 1 unit increase in \( x \). The y-intercept \( b = -42 \) indicates that the line crosses the y-axis at -42.

Point-Slope Form

The point-slope form of a linear equation is another useful method to represent a line, especially when you know the slope of the line and a specific point on the line. The formula is:\[ y - y_1 = m(x - x_1) \]Here:

• \( m \) is the slope of the line
• \( (x_1, y_1) \) are the coordinates of a known point on the line

This form is great for creating the equation of a line quickly, without needing to rearrange terms immediately. In our example, using the point (-8, -10) and the slope -4 (from the parallel line), we formed the equation: \( y + 10 = -4(x + 8) \). This demonstrates how the formula directly uses the slope and coordinates to define the line's behavior and position.

Parallel Lines

Parallel lines are unique in that they never intersect and always maintain the same separation distance apart. In geometry, understanding parallel lines is crucial for predicting the behavior of a line relative to another. The defining property of parallel lines is that they have identical slopes. This characteristic makes it simple to determine if two lines are parallel; you only need to compare their slopes.
In the exercise, the line that is parallel to \( y = -4x + 3 \) must also have a slope of -4, as parallel lines share the same slope. This concept allows us to quickly determine the slope for our new line equation that will pass through the point (-8, -10), ensuring it remains parallel to the given line. This principle is a powerful tool for solving problems involving lines in coordinate geometry.