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Chapter 5: Problem 36

Find the \(x\) -intercept and the \(y\) -intercept of the line with the given equation. Sketch the line using the intercepts. A calculator can be used to check the graph. $$y=0.25 x+4.5$$

Short Answer

Expert verified
Y-intercept is 4.5; x-intercept is -18. Plot (0, 4.5) and (-18, 0) and connect them with a line.

Step by step solution

01

Find the y-intercept

The y-intercept occurs where the line crosses the y-axis, which means that the value of x is 0. Substitute x=0 into the equation: \[ y = 0.25 \times 0 + 4.5 = 4.5 \]. Thus, the y-intercept is (0, 4.5).
02

Find the x-intercept

The x-intercept occurs where the line crosses the x-axis, which means that the value of y is 0. Set y=0 in the equation and solve for x: \[ 0 = 0.25x + 4.5 \]. Subtract 4.5 from both sides: \[ -4.5 = 0.25x \]. Now, divide by 0.25: \[ x = -18 \]. Thus, the x-intercept is (-18, 0).
03

Sketch the line using intercepts

To sketch the line, plot the points (0, 4.5) and (-18, 0) on a coordinate system. Draw a straight line passing through these points. This line represents the equation \( y = 0.25x + 4.5 \).

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Key Concepts

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

Finding Intercepts
Intercepts are the points where a line crosses the x-axis and y-axis on a graph. Understanding how to find these intercepts is crucial for graphing equations easily.

To find the y-intercept of a line, you need to determine where the line cuts across the y-axis. This happens when the value of x is zero since the y-axis is a vertical line running through x = 0. In the given equation, whenever you substitute x with zero, you calculate the y-intercept:
  • Set x = 0 in the equation: \[ y = 0.25 \times 0 + 4.5 = 4.5 \]
  • Thus, the y-intercept is at the point (0, 4.5).

For the x-intercept, you need to find out where the line crosses the x-axis, meaning y = 0. You plug y = 0 into the equation and solve for x:
  • Set y = 0 in the equation: \[ 0 = 0.25x + 4.5 \]
  • Solve for x: \[ 0.25x = -4.5 \]
  • Divide both sides by 0.25: \[ x = -18 \]
  • The x-intercept is at the point (-18, 0).
The intercepts help you to quickly visualize where a line is positioned on a graph.
Graphing Linear Equations
One of the methods to graph a linear equation is by using its intercepts, which simplifies the process significantly. The intercepts provide two vital points that can be easily plotted on a coordinate plane.

When graphing using intercepts:
  • First, find the x and y-intercepts as shown above.
  • Plot these points on the Cartesian plane - the y-intercept (0, 4.5) on the y-axis and the x-intercept (-18, 0) on the x-axis.
  • Connect these points with a straight line.
The line that you draw has to continue infinitely in both directions, passing precisely through these two intercept points. Using intercepts is efficient because calculating only two points is required, making it handy for sketching straight lines.
Coordinate Geometry
Coordinate geometry, also known as analytic geometry, involves using coordinates and algebra to interpret geometric problems. Graphing linear equations is a fundamental aspect of coordinate geometry, as it helps visually represent equation solutions.

In coordinate geometry:
  • The x-axis is the horizontal line, while the y-axis is the vertical line.
  • A point is represented as \((x, y)\) where "x" is the position along the x-axis and "y" is the position along the y-axis.
  • A linear equation represents a straight line across the coordinate plane.

The slope-intercept form of a line is given by the equation \(y = mx + b\), where "m" represents the slope of the line and "b" the y-intercept.
  • In our example, the equation \(y = 0.25x + 4.5\) tells us that 0.25 is the slope and 4.5 is the y-intercept.
Coordinate geometry allows us to analyze and interpret the properties of geometric figures, making it an essential tool for mathematical visualization.

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Most popular questions from this chapter

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Set up appropriate systems of equations. All numbers are accurate to at least two significant digits. An alloy used in electrical transformers contains nickel (Ni), iron (Fe), and molybdenum (Mo). The percent of Ni is \(1 \%\) less than five times the percent of Fe. The percent of Fe is \(1 \%\) more than three times the percent of Mo. Find the percent of each in the alloy.
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