Problem 39 - Algebra A Combined Function

Question

Write an equation of each line. Write the equation in standard form unless indicated otherwise. See Examples 1 through \(6 .\) Through (1,6) and (5,2)\(;\) use slope-intercept form.

Step-by-Step Solution

Answer

The equation in slope-intercept form is \( y = -x + 7 \).

Step 1: Determine the slope

To find the slope of a line passing through two points, use the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). For the points (1,6) and (5,2), plug in these values to get \( m = \frac{2 - 6}{5 - 1} = \frac{-4}{4} = -1 \).

Step 2: Use the slope-intercept form

The slope-intercept form of a line is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. We have found \( m = -1 \). Substitute this value into the equation to get \( y = -x + b \).

Step 3: Find the y-intercept

Use one of the given points to find \( b \). Using the point (1,6), substitute into \( y = mx + b \): \( 6 = -1(1) + b \). Solve for \( b \): \( 6 = -1 + b \), so \( b = 7 \).

Step 4: Write the final equation

Substitute \( b = 7 \) back into \( y = -x + b \) to get the equation of the line. Thus, the equation is \( y = -x + 7 \).

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is a common way to express the equation of a line. It is written as \( y = mx + b \), where:

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

This form is particularly useful because it directly shows the y-intercept and the slope, allowing you to easily graph the line. To convert from slope-intercept form to another form, such as the standard form, simply rearrange the terms accordingly.

Standard Form of a Line

The standard form of a line is typically expressed as \( Ax + By = C \), where \( A \), \( B \), and \( C \) are integers, and \( A \) should be a non-negative integer. An essential characteristic of standard form is that it allows us to analyze both x- and y-intercepts more straightforwardly.

• It emphasizes the coefficients of both variables in an equation.
• Useful for solving systems of linear equations, where you need the equations to have the variables aligned vertically.

To convert from slope-intercept form (\( y = mx + b \)) to standard form, you will rearrange the equation so that both \( x \) and \( y \) terms are on one side and the constant is on the other side. This involves moving terms using addition or subtraction and eliminating any fractional coefficients by multiplying through by a common factor.

Calculating Slope

The slope is a measure of the steepness and direction of a line, and it's a core concept when working with linear equations. To calculate the slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\), use the formula:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]

• If the slope is positive, the line rises as it moves from left to right.
• If the slope is negative, like in our problem (\( m = -1 \)), the line falls as it moves from left to right.
• A slope of zero indicates a horizontal line, while an undefined slope indicates a vertical line.

Calculating the slope accurately is key to forming the correct equation for a line, especially when you need to use the slope-intercept form or standard form. This process is also crucial for understanding how different lines interact and change in real-world applications.