Problem 53 - Algebra Form and Function

Question

Is the point-slope form or slopeintercept form the easier form to use when writing an equation for the line? Passes through (-5,10) and has slope 6

Step-by-Step Solution

Answer

Answer: The equation of the line is (y - 10) = 6(x + 5) in point-slope form, and y = 6x + 40 in slope-intercept form.

Step 1: Recall the point-slope and slope-intercept forms

The point-slope form of a line is given by (y - y1) = m(x - x1), where m is the slope and (x1, y1) is a point on the line. The slope-intercept form of a line is given by y = mx + b, where m is the slope and b is the y-intercept.

Step 2: Use the point-slope form

Given the point (-5, 10) and slope 6, we can substitute these values into the point-slope equation: (y - 10) = 6(x - (-5)) Simplify the equation: (y - 10) = 6(x + 5) This equation is now in point-slope form and we are done with this method.

Step 3: Use the slope-intercept form

To use the slope-intercept form, first, substitute the slope, m = 6, into the equation: y = 6x + b Next, use the given point (-5, 10) to solve for the y-intercept, b: 10 = 6(-5) + b Solve for b: 10 = -30 + b b = 40 Now, substitute the value of b back into the equation: y = 6x + 40 This equation is now in slope-intercept form and we are done with this method. #Conclusion# Comparing the two methods, we can observe that the point-slope form is easier to use in this case because it requires less calculation. We only need to plug in the given point and slope directly into the equation without solving for any additional variables. On the other hand, the slope-intercept form requires solving for the y-intercept, which adds an extra step to the process.

Key Concepts

Point-Slope Form

The point-slope form of a line is a straightforward way to express the equation of a line when you know a point on the line and its slope. This form is written as \( (y - y_1) = m(x - x_1) \), where \( m \) is the slope and \( (x_1, y_1) \) is a given point on the line. Using this form, you can quickly find the equation of a line without needing to calculate additional variables.
To use the point-slope form, simply:

• Identify the given point \( (x_1, y_1) \).
• Substitute \( x_1, y_1 \), and the slope \( m \) into the formula.
• Simplify, if necessary.

This method is particularly beneficial in situations where determining the y-intercept isn't necessary or when quick graphing is the goal. It keeps things simple by only requiring basic arithmetic operations directly in the equation.

Slope-Intercept Form

The slope-intercept form is another popular way of writing the equation of a line. This form is expressed as \( y = mx + b \), where \( m \) is the slope of the line, and \( b \) is the y-intercept. The y-intercept is the point where the line crosses the y-axis. Understanding both the slope and y-intercept allows you to sketch the line's path on a graph easily.To find the equation using this form:

• Substitute the given slope \( m \).
• Use a known point \( (x_1, y_1) \) to solve for \( b \), the y-intercept, by substituting the x and y values into the equation.
• Rearrange the equation to solve for \( b \).
• Write the complete equation with the determined \( m \) and \( b \).

This method provides the full expression of a line, making it useful when planning to graph lines, solving for y-intercept, or analyzing how lines intersect.

Graphing Lines

Graphing lines is a visual method to represent linear equations. It helps in understanding the relationship between variables and interpreting graphical data. Graphing can be accomplished using either the point-slope form or the slope-intercept form, each offering unique advantages.

• Using Point-Slope Form: With this form, plot the known point \( (x_1, y_1) \) on the graph. From this point, use the slope \( m \) to identify other points on the line by moving vertically and horizontally according to the slope's rise over run.
• Using Slope-Intercept Form: Start by plotting the y-intercept \( b \) on the y-axis. From the y-intercept, apply the slope \( m \) to locate other points on the line by moving according to the slope's ratio.

Graphing helps visualize the exact nature of lines, such as parallelism or orthogonality, and it aids in analyzing the real-world implications of these equations. Both forms lead to the same line, ensuring flexibility in approach based on the problem's context.