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Chapter 3: Problem 44

Use the intercept method to graph each equation. $$ 5 x-4 y=13 $$

Short Answer

Expert verified
The graph will cross the x-axis at \(\left(\frac{13}{5}, 0\right)\) and the y-axis at \(\left(0, -\frac{13}{4}\right)\).

Step by step solution

01

Understand the Intercept Method

The intercept method involves finding the points where the graph of an equation intersects the x-axis and y-axis. This means calculating where \(y=0\) (to find the x-intercept) and where \(x=0\) (to find the y-intercept).
02

Find the x-intercept

To find the x-intercept, set \(y = 0\) in the equation and solve for \(x\). Starting with \(5x - 4y = 13\), substitute \(y = 0\), giving \(5x = 13\). Solving for \(x\), we find \(x = \frac{13}{5}\). Thus, the x-intercept is at \(\left(\frac{13}{5}, 0\right)\).
03

Find the y-intercept

To find the y-intercept, set \(x = 0\) in the equation and solve for \(y\). Starting with \(5x - 4y = 13\), substitute \(x = 0\), giving \(-4y = 13\). Solving for \(y\), we find \(y = -\frac{13}{4}\). Thus, the y-intercept is at \(\left(0, -\frac{13}{4}\right)\).
04

Plot the Intercepts and Graph

Plot the points \(\left(\frac{13}{5}, 0\right)\) and \(\left(0, -\frac{13}{4}\right)\) on the graph. These are the intercepts where the line will cross the x-axis and y-axis. Draw a straight line through 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.

Intercept Method
Graphing linear equations using the intercept method is a straightforward and efficient way to visualize a line. It works by identifying the points where the line crosses the x-axis and y-axis. Here鈥檚 how it simplifies the process:
  • First, you need to find the x-intercept by setting the y value to zero in the equation and solving for x. This tells you where the line meets the x-axis.
  • Next, determine the y-intercept by doing the opposite鈥攕et x to zero and solve for y. This gives the point where the line hits the y-axis.
Once you have these two points, you can easily draw the line. There鈥檚 no need for additional calculations or complex algebra beyond these two steps. It鈥檚 a neat trick that saves time and reduces errors in plotting linear equations.
X-Intercept
The x-intercept of a line is the point where it crosses the x-axis. To find it when using the intercept method, you simply set the y variable to zero in your equation and solve for x. For our example equation, 5x - 4y = 13, when you substitute 0 for y, you鈥檙e left with 5x = 13. Solving for x gives you \(x = \frac{13}{5}\).
  • This means the line intercepts the x-axis at \(\left(\frac{13}{5}, 0\right)\).
  • It shows us where the graph is horizontally zero, a crucial point for constructing your graph.
Understanding the x-intercept is essential in graphing because it tells you where the line stops intersecting the horizontal plane. It鈥檚 also an excellent check-point to ensure your graphing is accurate and aligns with the calculated intersections.
Y-Intercept
The y-intercept is where the line crosses the y-axis. To find this intersection using the intercept method, you set x to zero in your equation and solve for y. Looking at our given equation, 5x - 4y = 13, replace x with 0 resulting in -4y = 13. Solving this gives you \(y = -\frac{13}{4}\).
  • This determines that the line intercepts the y-axis at \(\left(0, -\frac{13}{4}\right)\).
  • The y-intercept is helpful in graphing and confirms your line intercepts at that vertical point.
Identifying the y-intercept allows you to plot where the graph meets the vertical axis, a necessary step to sketch your equation correctly. Along with the x-intercept, it provides a foundational way to authenticate your graphical representation of a linear equation.

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