Problem 49 - Algebra 1: Concepts and Skills

Question

Write in standard form the equation of the line that passes through the given point and has the given slope. (Lesson 5.4 ) $$ (1,8), m=\frac{3}{4} $$

Step-by-Step Solution

Answer

The equation of the line in standard form that passes through the point (1,8) and has the slope of 3/4 is \(3x - 4y = -29\).

Step 1: Write the equation in point-slope form

Substitute the point (1,8) into the point-slope form \(y - y1 = m(x - x1)\). So the equation becomes \(y - 8 = \frac{3}{4}(x - 1)\).

Step 2: Simplify the equation

Use the distributive property \(\frac{3}{4} \times x - \frac{3}{4}\). So the equation becomes \(y - 8 = \frac{3}{4}x - \frac{3}{4}\). Then add 8 to both sides to isolate y. The equation now becomes \(y = \frac{3}{4}x + \frac{29}{4}\).

Step 3: Convert the equation to standard form

Multiply each term by 4 to get rid of the fraction. So the equation becomes \(4y = 3x + 29\). Now, subtract 3x from each side to get the standard form, \(3x - 4y = -29\).

Key Concepts

Standard Form of a Line

In algebra, the standard form of a linear equation is expressed as \( Ax + By = C \). In this form, \( A \), \( B \), and \( C \) are integers where \( A \) is non-negative, and \( A \), \( B \) cannot both be zero simultaneously. Standard form is particularly useful for easily determining an x-intercept or a y-intercept of a line by covering up parts of the equation. This form also makes it easier to work with systems of equations. In practice, gaining a clear view of how the coefficients relate to each other helps in better handling linear equations in more complex algebraic situations.

• The coefficient \( A \) should not be negative, often it is positive.
• The coefficients \( A \), \( B \), and \( C \) are typically reduced to the smallest integer that satisfies \( gcd(A, B, C) = 1 \).
• This form is notably different from other forms like slope-intercept and point-slope forms, which highlight different properties of lines.

Point-Slope Form

The point-slope form of a line is expressed through the formula: \( y - y_1 = m(x - x_1) \). This form uses a given slope \( m \) and a point \((x_1, y_1)\) that the line passes through. It is particularly helpful for writing the equation of a line when you know a point on the line and the slope.

• Why it's useful: It directly incorporates the slope and one specific point on the line, making it straightforward to derive the line's equation.
• How to use it: Plug the coordinates of the point into \((x_1, y_1)\), and the known slope into \(m\).
• Converting to other forms: From here, you can transform the equation into other forms such as slope-intercept or standard form as needed.

This form emphasizes the geometric way of understanding a line, clearly delineating the rise over run associated with the slope as well as its attachment to a particular point on the line.

Slope-Intercept Form

The slope-intercept form is one of the most commonly used ways to express the equation of a line. This form is written as \( y = mx + b \), where \( m \) is the slope of the line, and \( b \) is the y-intercept, representing the point where the line crosses the y-axis.

• Direct Interpretation: This form makes it easy to read the slope and y-intercept directly from the equation.
• Visualization: Provides a quick visual of how the line rises and falls across the graph and where it intersects the y-axis.
• Converting from point-slope form: Simple algebraic steps can convert a point-slope equation into slope-intercept for more intuitive reading.

When transitioning from another form, simply isolate \( y \) to get it into this format. This form is particularly appealing in scenarios where graphing is required, as it allows for fast plotting on standard Cartesian planes.

Distributive Property

The distributive property is a fundamental algebraic principle used extensively in simplifying expressions and equations. It is defined as \( a(b + c) = ab + ac \). In the context of line equations, it allows for simplifying the multiplication of terms such as applying a slope to variables inside parentheses.

• Application in equations: Helps eliminate parentheses by distributing the numerical term or variable across each term inside the parenthesis.
• Simplifying expressions: It is particularly useful when transferring equations from point-slope form to standard form or slope-intercept form, ensuring simpler or more precise representations of the line equation.
• Visualizing operations: By breaking down complex multiplications into simpler parts, it eases understanding and calculation.

Understanding and applying the distributive property streamlines working with linear equations, making it easier to handle transformations and solve for various unknowns in algebraic operations.