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Chapter 5: Problem 42

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

Short Answer

Expert verified
The slope is 1.

Step by step solution

01

- 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: \[ x = y + 5 \]
02

- Solve for \( y \)

Subtract \( 5 \) from both sides to solve for \( y \): \[ y = x - 5 \]
03

- 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, \( m = 1 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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鈥攖he 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鈥檚 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鈥檚 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.

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Most popular questions from this chapter

Sketch the graph of \(u-4 v=8\) using the horizontal axis for \(u\) values and the vertical axis for \(v\) values.

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