Problem 42 - Understanding Elementary Algebra with Geometry

Question

Determine the slope of the line from its equation. $$x-y=5$$

Step-by-Step Solution

Answer

The slope is 1.

Step 1 - Rewrite the equation in slope-intercept form

The slope-intercept form of a linear equation is given by: \[ y = mx + b \] where \( m \) is the slope and \( b \) is the y-intercept. Start by isolating \( y \) on one side of the equation. Add \( y \) to both sides: \[ x = y + 5 \]

Step 2 - Solve for \( y \)

Subtract \( 5 \) from both sides to solve for \( y \): \[ y = x - 5 \]

Step 3 - Identify the slope

Compare the equation \( y = x - 5 \) with the slope-intercept form \( y = mx + b \). The coefficient of \( x \) is the slope. Here, \( m = 1 \).

Key Concepts

linear equations

A linear equation is an equation that makes a straight line when graphed. It typically has the form: \[ Ax + By = C \]where \( A \), \( B \), and \( C \) are constants, and \( x \) and \( y \) are variables. Linear equations are fundamental in algebra and are used to represent many real-world situations.

• The line equation given in the problem is \( x - y = 5 \).
• To leverage this equation, we often convert it to an easier form for understanding—the slope-intercept form.

Understanding linear equations helps in finding points on the graph and identifying how variables relate to each other.

slope-intercept form

The slope-intercept form of a linear equation is one of the most intuitive ways to write and understand linear equations:\[ y = mx + b \]In this form:

• \( y \) is the value of the output variable.
• \( m \) is the slope of the line, indicating how steep the line is.
• \( x \) is the value of the input variable.
• \( b \) is the y-intercept, the point where the line crosses the y-axis.

Let’s convert the given equation \( x - y = 5 \) into slope-intercept form to find the slope.

solving for y

Solving for \( y \) involves rearranging the original equation so that \( y \) is isolated on one side. Let’s solve the given equation step by step:

• Original equation: \( x - y = 5 \).
• Add \( y \) to both sides: \( x = y + 5 \).
• Subtract \( 5 \) from both sides to isolate \( y \): \( y = x - 5 \).

Now we have the equation in slope-intercept form (\( y = mx + b \)):

• \( y = x - 5 \)

By comparing this with the slope-intercept form \( y = mx + b \), we find the slope (\( m \)) to be \( 1 \). This means for each unit increase in \( x \), \( y \) increases by 1.