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Chapter 3: Problem 25

Find the intercepts and graph them. $$ -12 x+13 y=1 $$

Short Answer

Expert verified
X-intercept: \(-\frac{1}{12}\), Y-intercept: \(\frac{1}{13}\); graph is a straight line through these intercepts.

Step by step solution

01

Find the x-intercept

The x-intercept occurs when \( y = 0 \). Substitute \( y = 0 \) into the equation \(-12x + 13y = 1\) to find \( x \).\(-12x + 13(0) = 1\)\(-12x = 1\)\(x = -\frac{1}{12}\).So, the x-intercept is \( \left(-\frac{1}{12}, 0\right) \).
02

Find the y-intercept

The y-intercept occurs when \( x = 0 \). Substitute \( x = 0 \) into the equation \(-12x + 13y = 1\) to find \( y \).\(-12(0) + 13y = 1\)\(13y = 1\)\(y = \frac{1}{13}\).So, the y-intercept is \( \left(0, \frac{1}{13}\right) \).
03

Plot the intercepts on a graph

To graph the line, plot the intercepts \( \left(-\frac{1}{12}, 0\right) \) and \( \left(0, \frac{1}{13}\right) \) on a coordinate plane. These points are where the line crosses the x-axis and y-axis respectively. Draw a straight line connecting these two points to complete the graph of the equation.

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Key Concepts

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

Intercepts
An intercept is a point where a line crosses one of the axes on the coordinate plane. In this case, we're dealing with a linear equation, which means it has two types of intercepts: x-intercept and y-intercept.
  • X-Intercept: This occurs where the line crosses the x-axis. At the x-intercept, the y-coordinate is always zero. To find the x-intercept, substitute zero for y in the equation, and solve for x. For the equation \(-12x + 13y = 1\), setting \(y = 0\) gives \(x = -\frac{1}{12}\). Hence, the x-intercept is \((-\frac{1}{12}, 0)\).
  • Y-Intercept: This is where the line intersects the y-axis. At this point, the x-coordinate is zero. To determine the y-intercept, substitute zero for x and solve the equation for y. For our equation, using \(x = 0\), \(-12x + 13y = 1\) simplifies to \((0, \frac{1}{13})\).
These intercepts are crucial as they provide specific points that help in sketching the graph of the linear equation.
Graphing
Graphing is a method of visually representing a linear equation on a coordinate plane by plotting points and drawing a line through these points. It's an essential aspect of understanding how equations behave and interact.
Here's how you would graph the linear equation \(-12x + 13y = 1\):
  • Identify Intercepts: Begin by finding both the x-intercept and y-intercept. In this case, the x-intercept is \((-\frac{1}{12}, 0)\) and the y-intercept is \((0, \frac{1}{13})\).
  • Plot Intercepts: On the coordinate plane, mark the intercept points. These points will serve as reference points for your line.
  • Draw the Line: Connect these points with a straight line. Since this is a straight line equation, it continues infinitely in both directions, but for graphical purposes, you will draw it across your graph space.
Graphing reinforces the visual understanding of the relationship between x and y as depicted by the equation.
Coordinate Plane
The coordinate plane is a two-dimensional surface where points are defined using pairs of numbers, known as coordinates. It consists of two perpendicular lines, the horizontal x-axis and the vertical y-axis, which intersect at a central point called the origin \(0, 0\).
In the context of graphing a linear equation like \(-12x + 13y = 1\):
  • Understanding the Axes: The x-axis extends horizontally and represents the domain of x-values. The y-axis is vertical and represents the range of y-values.
  • Plotting Points: Each point on the coordinate plane is identified by an ordered pair \(x, y\). For example, the point \((-\frac{1}{12}, 0)\) is positioned left on the x-axis, and \(0, \frac{1}{13})\) is up on the y-axis.
  • Quadrants: The coordinate plane is divided into four quadrants based on the signs of the x and y coordinates. This helps in situating points precisely when graphing.
The coordinate plane serves as the canvas on which lines and curves formed by equations can be sketched, offering a tangible form to abstract algebraic expressions.

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