Problem 45 - Algebra A Combined Function

Question

Write an equation of each line. Write the equation in standard form unless indicated otherwise. See Examples 1 through \(6 .\) Slope \(-\frac{4}{3} ;\) through (-5,0)

Step-by-Step Solution

Answer

The equation of the line is \(4x + 3y = -20\) in standard form.

Step 1: Understand the Formula for a Line

The formula for a line in point-slope form is \( y - y_1 = m(x - x1) \), where \( m \) is the slope, and \((x_1, y_1)\) is a point on the line.

Step 2: Plug in the Known Values

Using the given point \((-5, 0)\) and slope \(-\frac{4}{3}\), substitute these into the point-slope formula. This gives: \( y - 0 = -\frac{4}{3}(x + 5) \).

Step 3: Distribute the Slope

Expand the equation by distributing the slope across the parentheses: \( y = -\frac{4}{3}x -\frac{4}{3} imes 5 \).

Step 4: Simplify the Equation

Calculate \(-\frac{4}{3} \times 5\) to get \(-\frac{20}{3}\), so the equation becomes: \( y = -\frac{4}{3}x - \frac{20}{3} \).

Step 5: Convert to Standard Form

Multiply every term by 3 to eliminate the fractions: \( 3y = -4x - 20 \).

Step 6: Rearrange into Standard Form

Move all terms to one side to get the equation in standard form: \( 4x + 3y = -20 \).

Key Concepts

Slope-Intercept Form

The slope-intercept form is arguably the most popular way to express the equation of a line. It is written as \( y = mx + b \), where \( m \) represents the slope of the line and \( b \) is the y-intercept. This form makes it easy to identify the slope and the value where the line crosses the y-axis. Understanding the slope-intercept form is essential because it visually represents how much the line rises or falls for each unit it moves horizontally.

Key points to remember about slope-intercept form:

• \( m \) is the slope, calculated as the change in y over the change in x.
• \( b \) is where the line intersects the y-axis, allowing you to quickly graph the line starting from this point.

If you know the slope and y-intercept of a line, you can swiftly write its equation using this straightforward form. As seen in our exercise, the line can first be represented in point-slope form before easily converting to slope-intercept form by isolating \( y \).

Standard Form

The standard form of a line equation aims to rid the equation of fractions and neatly categorizes each term. It is typically structured as \( Ax + By = C \), where \( A \), \( B \), and \( C \) are integers, and \( A \) should be positive if possible. This form is especially useful when you need to analyze or compare multiple equations. Working with integers rather than fractions simplifies arithmetic operations significantly.

To convert an equation to standard form, follow these steps:

• Ensure all terms involving variables are on one side of the equation.
• Multiply through by a common denominator if fractions are present to clear them.
• Adjust the equation so that \( A \) is positive by multiplying all terms by \(-1\) if needed.

In our exercise, this form was the final step, ensuring clarity and ease of interpretation. It is helpful in contexts such as comparing equations or systems of equations.

Point-Slope Form

The point-slope form is another practical way to write the equation of a line, especially useful when you know a specific point and the slope. It is 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 effectively sets a foundation for shifting between other forms like slope-intercept or standard forms.

This form is advantageous because:

• You can immediately apply it with any known point and slope to draft the beginning of a line equation.
• It provides a straightforward method for encoding line information when starting with less complete information than other forms might require.

In the exercise example, point-slope form was the starting point to illustrate the entire equation development process, serving as a bridge to subsequent transformations into other forms.