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Chapter 3: Problem 52

Find the intercepts. Then graph. $$ x-1=y $$

Short Answer

Expert verified
Y-intercept: (0, -1), X-intercept: (1, 0). Graph has a line through these points.

Step by step solution

01

Rewrite the Equation

Rewrite the given equation in the form of the line equation by solving for y: \( y = x - 1 \).
02

Find the Y-Intercept

To find the y-intercept, set \( x = 0 \) and solve for \( y \): \[ y = 0 - 1 \] Therefore, \( y = -1 \). The y-intercept is at \( (0, -1) \).
03

Find the X-Intercept

To find the x-intercept, set \( y = 0 \) and solve for \( x \): \[ 0 = x - 1 \] Therefore, \( x = 1 \). The x-intercept is at \( (1, 0) \).
04

Plot Intercepts and Draw the Line

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

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Key Concepts

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

graphing linear equations
Graphing linear equations is a core skill in algebra that helps us understand relationships between two variables. Linear equations form straight lines when graphed. A standard form of a linear equation is given by Ax + By = C , but it can also be written as y = mx + b, where m is the slope and b is the y-intercept. To graph a linear equation, follow these steps:
  • Rewrite the equation, if necessary, to express y in terms of x.
  • Find the x-intercept and y-intercept.
  • Plot the intercepts on the graph.
  • Draw a line through the points.
A linear equation has infinitely many solutions that all lie on the line. Each solution represents a point on the graph. Graphing linear equations helps visualize the solutions and understand how changes in the equation affect the line's slope and position.
x-intercept
The x-intercept of a linear equation is the point where the graph crosses the x-axis. At this point, y equals zero. To find the x-intercept, we set y to zero in the equation and solve for x. In the example, starting from: y = x - 1
  • Set y to zero: 0 = x - 1
  • Solve for x: x = 1
The x-intercept is at (1,0). This means the line crosses the x-axis at x=1. We often plot this point on the graph first, along with the y-intercept, to easily draw the line representing the equation.
y-intercept
The y-intercept is where the graph of an equation intersects the y-axis. At this point, x equals zero. To find the y-intercept of a linear equation, set x to zero and solve for y. Using our example equation: y = x - 1
  • Set x to zero: y = 0 - 1
  • Solve for y: y = -1
Thus, the y-intercept is at (0, -1). This indicates the graph crosses the y-axis at y=-1. Plotting both the x-intercept and y-intercept on the graph helps in accurately drawing the line representing the equation. Remember, the y-intercept is the point (0, b) in the slope-intercept form y=mx+b.
line equations
Linear equations, often called line equations, are algebraic expressions that form straight lines when graphed. The most commonly used form is y = mx + b, where:
  • m is the slope, indicating the steepness of the line.
  • b is the y-intercept, indicating where the line crosses the y-axis.
In the equation y = x - 1:
  • The slope (m) is 1, meaning the line rises one unit for every unit it moves to the right.
  • The y-intercept (b) is -1, meaning the line crosses the y-axis at (0, -1).
By understanding line equations, we can predict and plot the behavior of linear relationships. Graphing these equations involves finding intercepts, plotting them, and drawing the line that best represents the equation's solutions over the entire plane.

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