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Chapter 5: Problem 125

Graph \(3 x-2 y=6\) using intercepts. (Section 3.2 Example 4 )

Short Answer

Expert verified
The x-intercept and y-intercept are 2 and 3 respectively. This is used to graph the linear equation.

Step by step solution

01

Determining the x-intercept

To find the x-intercept, set y = 0 and solve for x in the given equation \(3 x - 2 y = 6\). Therefore \(x = \frac{6}{3} = 2\). So, the x-intercept is 2.
02

Determining the y-intercept

To find the y-intercept, set x = 0 in the given equation \(3 x - 2 y = 6\). Then, solve for y to get \(y = \frac{-6}{-2} = 3\). The y-intercept is 3.
03

Graphing the Equation

Now, plot these intercepts on the Cartesian plane. This gives two points (2,0) for the x-intercept and (0,3) for the y-intercept. Draw a straight line passing through these points. This line represents the graph of the given linear equation.

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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 is a fundamental concept in graphing linear equations. It is the point where the graph crosses the x-axis on a Cartesian plane. This means that at the x-intercept, the value of y is always zero.
To find the x-intercept of a linear equation, substitute 0 for y in the equation, and solve for x. For example, if we have the equation \(3x - 2y = 6\), setting \(y = 0\) gives us \(3x = 6\). Solving this equation, we find \(x = 2\). Hence, the x-intercept is 2.
Remember, knowing the x-intercept helps you understand where the line hits the x-axis, crucial for graphing.
y-intercept
The y-intercept is the point where a graph crosses the y-axis. At this point, the value of x is always zero. This is important in graphing because it tells us one of the limits on the vertical scale.
To calculate the y-intercept, replace x with 0 in the equation and solve for y. Using the equation \(3x - 2y = 6\), if \(x = 0\), we simplify to \(-2y = 6\). Solving for y, we get \(y = 3\). Therefore, the y-intercept is 3.
This straightforward method shows us exactly where a line crosses the y-axis, aiding in accurately plotting the linear equation.
Cartesian plane
The Cartesian plane, named after Ren茅 Descartes, is a vital tool in algebra. It consists of two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). These axes divide the plane into four quadrants and are used to plot points through ordered pairs (x, y).
In our linear equation example, we use the Cartesian plane to plot intercepts. Each axis serves as a guideline to help determine the position of these intercepts. The x-axis lies where \(y = 0\) and the y-axis lies where \(x = 0\).
This plane is essential for visualizing how algebraic solutions fit into the 'real world,' making complex data easy to interpret visually.
linear equation
A linear equation is an equation between two variables that gives a straight line when plotted on a graph. It's usually in the form \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants. Graphing a linear equation shows the relationship between x and y, clearly depicted as a line.
In our case, \(3x - 2y = 6\) is linear. By finding intercepts and plotting on the Cartesian plane, the relationship is expressed visually. The slope of this line doesn't change, reflecting a consistent rate of increase or decrease.
Understanding linear equations allows you to precisely describe how variables interact, enabling predictions and inferences in mathematical modeling.

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Most popular questions from this chapter

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. I used the product rule for exponents to multiply \(x^{7}\) and \(y^{9}\)

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