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Chapter 4: Problem 187

Graph using the intercepts. \(x-y=1\)

Short Answer

Expert verified
Plot points (1, 0) and (0, -1), then draw a line through them.

Step by step solution

01

Find the x-intercept

To find the x-intercept, set y to 0 and solve for x. \( x - y = 1 \) becomes \( x - 0 = 1 \), so \( x = 1 \). Thus, the x-intercept is at (1, 0).
02

Find the y-intercept

To find the y-intercept, set x to 0 and solve for y. \( x - y = 1 \) becomes \( 0 - y = 1 \), so \( -y = 1 \) and \( y = -1 \). Thus, the y-intercept is at (0, -1).
03

Plot the intercepts

Plot the points (1, 0) and (0, -1) on the coordinate plane. These points are the x and y intercepts.
04

Draw the line

Draw a straight line through the points (1, 0) and (0, -1). This line represents the graph of the equation \( x - y = 1 \).

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Key Concepts

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

x-intercept
The x-intercept of a linear equation is the point where the line crosses the x-axis. To find the x-intercept, you set the y-value to zero and solve for x. For example, in the equation \(x - y = 1\), set \(y = 0\). This simplifies the equation to \(x - 0 = 1\), thus \(x = 1\). So, the x-intercept is at the point (1, 0). It's important to understand this because it provides a crucial point from which you can start graphing the line.
y-intercept
The y-intercept is where the line crosses the y-axis. To determine the y-intercept, you set the x-value to zero and solve for y. Using the same example, \(x - y = 1\), set \(x = 0\). This turns the equation into \(0 - y = 1\), which simplifies to \(-y = 1\). Solving for y gives \(y = -1\). So, the y-intercept is at (0, -1). This point also helps in accurately drawing the graph of the equation.
coordinate plane
The coordinate plane is a two-dimensional surface where we can graph equations. It consists of a horizontal axis (x-axis) and a vertical axis (y-axis). Each point on the plane is defined by a pair of numerical coordinates, such as (1,0) or (0,-1). These coordinates represent the values of x and y at that point. Plotting points like the x-intercept and y-intercept on the coordinate plane helps visualize the line represented by the equation.

For instance, plotting (1, 0) and (0, -1) on the plane helps us see that our line, \(x - y = 1\), starts at these intercepts.
solving equations
Solving equations is a fundamental skill in algebra that involves finding the value of the variables that satisfy the equation. To graph a linear equation using the intercepts, you solve for x to find the x-intercept and solve for y to find the y-intercept.

Using our example \(x - y = 1\), setting \(y = 0\) helped us solve for \(x = 1\) to find the x-intercept. Similarly, setting \(x = 0\) allowed us to solve for \(y = -1\) for the y-intercept.
  • This method simplifies graphing as it uses simple algebraic manipulation,
  • It leverages the intercepts as key points to get an accurate line on the graph.

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