Problem 2 - Elementary Algebra

Question

For Problems 1-12, find the equation of the line that contains the given point and has the given slope. Express equations in the form \(A x+B y=C\), where \(A, B\), and \(C\) are integers. $$ (5,2), m=\frac{3}{7} $$

Step-by-Step Solution

Answer

The equation of the line is \(3x - 7y = 1\).

Step 1: Understand the Line Equation Form

We need to express the equation of the line in the form \(Ax + By = C\) given a point \((x_1, y_1) = (5, 2)\) and a slope \(m = \frac{3}{7}\). This is a line in standard form.

Step 2: Use the Point-Slope Form

Start with the point-slope form of the equation, which is \(y - y_1 = m(x - x_1)\). Substitute \(x_1 = 5\), \(y_1 = 2\), and \(m = \frac{3}{7}\) into the formula, so it becomes \(y - 2 = \frac{3}{7}(x - 5)\).

Step 3: Distribute the Slope

Apply the distributive property to the right side: \(y - 2 = \frac{3}{7}x - \frac{15}{7}\).

Step 4: Eliminate the Fraction

Multiply every term by 7 to eliminate the fraction, yielding \(7y - 14 = 3x - 15\).

Step 5: Rearrange to Standard Form

Rearrange the equation to achieve the form \(Ax + By = C\). Subtract 3x from both sides to get \(-3x + 7y = -1\).

Step 6: Standardize Signs

Multiply the entire equation by -1 to remove the negative sign from \(A\), resulting in \(3x - 7y = 1\).

Key Concepts

Point-Slope Form

The point-slope form of a line is a handy tool for finding the equation of a line when you know one point it passes through and its slope. Point-slope form is written as: \( y - y_1 = m(x - x_1) \), where:

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

This form is particularly useful because it directly incorporates both a point on the line and the slope, making it straightforward to set up an equation.

To apply this form, simply substitute the known point and slope into the formula. For example, with the point \( (5, 2) \) and the slope \( \frac{3}{7} \), the equation becomes \( y - 2 = \frac{3}{7}(x - 5) \). From there, you can further manipulate the equation into other forms if needed.

Slope-Intercept Form

Once you have an equation in the point-slope form, transforming it into the slope-intercept form can sometimes provide a clearer picture of the line's behavior. The slope-intercept form is expressed as: \( y = mx + b \), where:

• \( m \) is the slope of the line.
• \( b \) is the y-intercept, the point where the line crosses the y-axis.

To change from the point-slope form \( y - 2 = \frac{3}{7}(x - 5) \) to this form, distribute the slope on the right side and solve for \( y \).

You will first calculate: \( y - 2 = \frac{3}{7}x - \frac{15}{7} \) and then add 2 to both sides to solve a bit further to reveal \( y = \frac{3}{7}x + \frac{-1}{7} \). Now \( y \) is isolated, allowing you to see how the line behaves across different x values.

Standard Form of a Line

The standard form of a line brings the equation into a neat package with integers, often preferred for different types of calculations or geometrical interpretations. The standard form is expressed as: \( Ax + By = C \), where:

• \( A, B, \) and \( C \) are integers.
• \( A \) should ideally be a positive number.

Transforming our equation \( y = \frac{3}{7}x + \frac{-1}{7} \) into the standard form involves clearing fractions and rearranging terms.

First multiply through by 7 to eliminate fractions: \( 7y = 3x - 1 \).Rearrange to bring \( x \) and \( y \) terms to one side: \( -3x + 7y = -1 \). To finalize, multiply all terms by \(-1\) to achieve the standard form with \( A \) as a positive integer: \( 3x - 7y = 1 \). This format is particularly useful when you need to verify perpendicularity or parallelism of lines.