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Chapter 3: Problem 54

Find the \(x\) - and \(y\) -intercepts for each line and use them to graph the line. $$2 x+3 y=30$$

Short Answer

Expert verified
x-intercept: (15, 0), y-intercept: (0, 10). Plot these points and draw a line through them.

Step by step solution

01

Find the x-intercept

To find the x-intercept, set the value of y to 0 in the equation. Start with the given equation: \[2x + 3y = 30\]Next, set y to 0:\[2x + 3(0) = 30\]This simplifies to:\[2x = 30\]Solving for x, we get:\[x = 15\]Therefore, the x-intercept is (15, 0).
02

Find the y-intercept

To find the y-intercept, set the value of x to 0 in the equation. Start with the given equation: \[2x + 3y = 30\]Next, set x to 0:\[2(0) + 3y = 30\]This simplifies to:\[3y = 30\]Solving for y, we get:\[y = 10\]Therefore, the y-intercept is (0, 10).
03

Plot the intercepts on the graph

Now that we have the intercepts (15, 0) and (0, 10), plot these points on a graph. The x-intercept (15, 0) should be placed at point where the x-axis is 15. The y-intercept (0, 10) should be placed at point where the y-axis is 10.
04

Draw the line through the intercepts

Draw a straight line through the points (15, 0) and (0, 10) on the graph. This line represents the equation \[2x + 3y = 30\].

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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 line is the point where the line crosses the x-axis. This point is always of the form \((x, 0)\) because on the x-axis, the y-coordinate is always 0. To find the x-intercept for the equation \[2x + 3y = 30\], you set \(y = 0\) and solve for \(x\).

Start with the equation:
\[2x + 3y = 30\]
Set \(y = 0\):
\[2x + 3(0) = 30\]
Simplify:
\[2x = 30\]
Solve for \(x\):
\[x = 15\]
Therefore, the x-intercept is \( (15, 0) \).
  • x-intercept is where the line meets the x-axis
  • Found by setting \(y \) to 0
  • Solved by simplifying and isolating \( x \)
y-intercept
The y-intercept is the point where the line crosses the y-axis. This point is always of the form \((0, y)\) because on the y-axis, the x-coordinate is always 0. To find the y-intercept for the equation \[2x + 3y = 30\], you set \(x = 0\) and solve for \(y\).

Start with the same equation:
\[2x + 3y = 30\]
Set \(x = 0\):
\[2(0) + 3y = 30\]
Simplify:
\[3y = 30\]
Solve for \(y\):
\[y = 10\]
Therefore, the y-intercept is \( (0, 10) \).
  • y-intercept is where the line meets the y-axis
  • Found by setting \( x \) to 0
  • Solved by simplifying and isolating \( y \)
plotting points
After finding the x-intercept and y-intercept, you can plot these points on a graph to draw the line. These intercepts are key because they give you two precise points through which the line passes.

First, locate the x-intercept (15, 0) on the graph. This will be a point on the x-axis, 15 units to the right of the origin.

Next, locate the y-intercept (0, 10). This will be a point on the y-axis, 10 units up from the origin.
  • x-intercept gives a point where the line crosses the x-axis
  • y-intercept gives a point where the line crosses the y-axis
  • Plotting these points helps in visualizing where the line is on the graph

Once you have these points plotted, draw a straight line passing through them. This line is the graphical representation of the equation \[2x + 3y = 30\].

  • Start by plotting the intercepts
  • Use a ruler to draw a straight line through these points
  • This visual method helps in understanding the relationship between the coordinates and the equation

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