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Chapter 11: Problem 145

In the following exercises, find the intercepts. $$ 3 x-2 y=12 $$

Short Answer

Expert verified
The intercepts are (4, 0) and (0, -6).

Step by step solution

01

Find the x-intercept

To find the x-intercept, set y = 0 in the equation and solve for x. Start with the equation: \[ 3x - 2y = 12 \] Substitute y = 0: \[ 3x - 2(0) = 12 \] This simplifies to: \[ 3x = 12 \] Now, divide both sides by 3: \[ x = 4 \]So, the x-intercept is (4, 0).
02

Find the y-intercept

To find the y-intercept, set x = 0 in the equation and solve for y. Start with the equation: \[ 3x - 2y = 12 \] Substitute x = 0: \[ 3(0) - 2y = 12 \] This simplifies to: \[ -2y = 12 \] Now, divide both sides by -2: \[ y = -6 \]So, the y-intercept is (0, -6).

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Key Concepts

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

Finding the x-intercept
To find the x-intercept of a linear equation, you need to determine where the line crosses the x-axis. This occurs when the y-coordinate is zero. The process involves substituting y = 0 into the equation. For example, consider the equation \[ 3x - 2y = 12 \]. By substituting y = 0 into the equation, it becomes \[ 3x - 2(0) = 12 \]. Simplifying this, we get \[ 3x = 12 \]. Dividing both sides by 3, we get \[ x = 4 \]. Therefore, the x-intercept is (4, 0). This means the line crosses the x-axis at the point (4, 0).
Finding the y-intercept
The y-intercept of a linear equation is the point where the line crosses the y-axis. This happens when the x-coordinate is zero. To find the y-intercept, you substitute x = 0 into the equation. Let's use the equation \[ 3x - 2y = 12 \] for example. By substituting x = 0, the equation simplifies to \[ 3(0) - 2y = 12 \]. This further simplifies to \[ -2y = 12 \]. Dividing both sides by -2, we get \[ y = -6 \]. So, the y-intercept is (0, -6). This means the line crosses the y-axis at the point (0, -6).
Understanding Linear Equations
Linear equations represent straight lines in a coordinate plane. They generally have the form \[ Ax + By = C \], where A, B, and C are constants. The solutions to these equations are ordered pairs (x, y) that make the equation true. For instance, in the equation \[ 3x - 2y = 12 \], A = 3, B = -2, and C = 12.
The graph of this equation is a straight line where each point (x, y) on the line satisfies the equation. The x-intercept and y-intercept are specific solutions that occur at the points where the line crosses the x-axis and y-axis, respectively. Understanding these intercepts can help in graphing the equation and understanding the relation between the variables.

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Most popular questions from this chapter

What does the sign of the slope tell you about a line?

Graph the line passing through \((1,1)\) with slope \(m=\frac{3}{2}\)

In the following exercises, graph the vertical and horizontal lines. $$ y=-5 $$

In the following exercises, for each ordered pair, decide a. if the ordered pair is a solution to the equation. b. if the point is on the line. \(y=\frac{2}{3} x-1\) (a) \((0,-1)\) (b) \((3,1)\) (c) \((-3,-3)\) (d) \((6,4)\)

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