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Chapter 3: Problem 3

Determine the slope and \(y\) -intercept. $$ x-y=5 $$

Short Answer

Expert verified
Slope = 1, Y-intercept = -5.

Step by step solution

01

Rewrite the Equation in Slope-Intercept Form

The slope-intercept form of a line is given by the equation \( y = mx + b \), where \( m \) is the slope and \( b \) is the \( y \)-intercept. Start by solving the given equation \( x - y = 5 \) for \( y \).\[ x - y = 5 \]Subtract \( x \) from both sides:\[ -y = -x + 5 \]Multiply each term by \( -1 \) to solve for \( y \):\[ y = x - 5 \]
02

Identify the Slope and Y-Intercept

Once the equation is in the form \( y = mx + b \), identify the slope \( m \) and the \( y \)-intercept \( b \). From the equation \( y = x - 5 \), you can see that:- The slope \( m \) is the coefficient of \( x \), which is \( 1 \).- The \( y \)-intercept \( b \) is the constant term, which is \( -5 \).

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Key Concepts

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

Slope
The slope of a line is a measure of its steepness or inclination. You can think of it as how much the line rises or falls vertically for each unit it moves horizontally. In mathematical terms, the slope is represented by the letter \( m \). This concept is crucial when dealing with linear equations, as it tells you how the values of \( y \) change relative to the values of \( x \).
  • A positive slope means the line is ascending as you move from left to right.
  • A negative slope means the line is descending.
  • If the slope is zero, the line is horizontal.
  • If the slope is undefined, the line is vertical.
In our example, the linear equation was rewritten as \( y = x - 5 \). Here, the slope \( m \) is 1, indicating that for every 1 unit increase in \( x \), \( y \) also increases by 1 unit.
Y-Intercept
The \( y \)-intercept of a line is the point where the line crosses the \( y \)-axis. In the slope-intercept form \( y = mx + b \), the \( y \)-intercept is denoted by \( b \). It provides a starting point for the graph of the line on the Cartesian plane.
  • The \( y \)-intercept is valuable for quickly sketching the line, as it tells you where to place the first point on the \( y \)-axis.
  • Knowing the \( y \)-intercept also helps in understanding the shift of the line up or down on the graph.
For the equation \( y = x - 5 \), the \( y \)-intercept is \( -5 \). This means the line crosses the \( y \)-axis at the point (0, -5). It informs us that when \( x = 0 \), \( y \) is -5.
Linear Equation
A linear equation is a mathematical statement that describes a straight line on a graph. The most common form of a linear equation is the slope-intercept form \( y = mx + b \), where the graph of this equation is a line. Linear equations are fundamental in algebra and are useful for modeling various real-world relationships. They are characterized by:
  • Having variables raised to the power of 1.
  • Producing a straight line when graphed.
  • Being used to represent proportional relationships.
In our task, we started with the equation \( x - y = 5 \). By rearranging it to \( y = x - 5 \), we successfully expressed it in the slope-intercept form, which highlights both the slope and \( y \)-intercept clearly. Recognizing and rewriting equations in this form makes it easier to interpret and graph them.

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Most popular questions from this chapter

Find at least five ordered pair solutions and graph. $$ x=-3 $$

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