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Chapter 2: Problem 40

Graph each equation using the intercept method. Label the intercepts on each graph. \(-5 x+3 y=11\)

Short Answer

Expert verified
X-intercept is \( \left( \frac{-11}{5}, 0 \right) \) and y-intercept is \( \left( 0, \frac{11}{3} \right) \).

Step by step solution

01

Find the x-intercept

To find the x-intercept, set the value of y to 0 in the equation and solve for x. The given equation is \[-5x + 3y = 11\].By setting \(y = 0\), it becomes:\[-5x + 3(0) = 11\]\[-5x = 11\]\[x = \frac{-11}{5}\].Thus, the x-intercept is \( \left( \frac{-11}{5}, 0 \right) \).
02

Find the y-intercept

To find the y-intercept, set the value of x to 0 in the equation and solve for y. Using the equation \[-5x + 3y = 11\], set \(x = 0\):\[-5(0) + 3y = 11\]\[3y = 11\]\[y = \frac{11}{3}\].Thus, the y-intercept is \( (0, \frac{11}{3} ) \).
03

Plot the intercepts

On a graph, plot the two intercepts:- The x-intercept \( \left( \frac{-11}{5}, 0 \right) \) on the x-axis.- The y-intercept \( \left( 0, \frac{11}{3} \right) \) on the y-axis.
04

Draw the line

Draw a straight line through the points \( \left( \frac{-11}{5}, 0 \right) \) and \( \left( 0, \frac{11}{3} \right) \). This line represents the equation \(-5x + 3y = 11\).
05

Label the Intercepts

Label the intercepts on the graph clearly as \( \left( \frac{-11}{5}, 0 \right) \) for the x-intercept and \( \left( 0, \frac{11}{3} \right) \) for the y-intercept to validate the plotted points.

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Key Concepts

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

Intercept Method
The intercept method is a simple way to graph linear equations. This method involves finding both the x-intercept and y-intercept of a line. These intercepts are key points where our line will intersect the x-axis and the y-axis, respectively.
Once we have these points, we can draw a straight line through them to represent the solution to the equation. This technique is particularly useful because it only requires two points to graph a line accurately.
  • First, find the x-intercept by setting y to 0 in the equation.
  • Next, find the y-intercept by setting x to 0.
  • Finally, plot these intercepts on a graph and draw the line through them.
By labeling our graph with these intercepts, it makes it easy to verify our work and ensure the proper line has been plotted.
X-Intercept
The x-intercept is the point where a line crosses the x-axis. This means that the value of y is 0 at this point. To find the x-intercept, set y to 0 in the equation and solve for x.
For the equation \(-5x + 3y = 11\), we make y equal to zero to get \(-5x = 11\). Solving this gives us the x-intercept as \(x = \frac{-11}{5}\).
The coordinates for the x-intercept are therefore \( \left( \frac{-11}{5}, 0 \right) \). This gives us a clear point to plot on the x-axis, which is crucial for the intercept method.
Visualizing this point will help in drawing the entire line.
Y-Intercept
The y-intercept describes where a line touches the y-axis. Here, the value of x is 0. To find the y-intercept, set x to 0 in the equation and solve for y.
For example, with the equation \(-5x + 3y = 11\), setting x as zero changes it to \(3y = 11\). Solving this gives \(y = \frac{11}{3}\).
So, the y-intercept is \( \left( 0, \frac{11}{3} \right)\), a point we can precisely plot on the y-axis.
Using both the x and y intercepts allows us to establish a line and visualize how the equation behaves.
Identifying these intercepts is fundamental for the accurate graphing of linear equations.
Plotting Points
Plotting points on a graph is the critical step in visualizing linear equations. This involves marking specific coordinates that have been identified, such as the x-intercept and y-intercept.
In our example, the intercepts were determined as \( \left( \frac{-11}{5}, 0 \right)\) and \( \left( 0, \frac{11}{3} \right)\). To correctly plot these:
  • First, position the x-intercept on the x-axis at \( \frac{-11}{5} \).
  • Secondly, indicate the y-intercept on the y-axis at \( \frac{11}{3} \).

Once these points are plotted, draw a straight line through them to portray the solution for \(-5x + 3y = 11\). This graphical representation provides clarity and insight into the equation.

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