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Chapter 3: Problem 40

Find the intercepts and graph them. $$ 3 x+12 y=-4 $$

Short Answer

Expert verified
The x-intercept is \(-\frac{4}{3}\), and the y-intercept is \(-\frac{1}{3}\).

Step by step solution

01

Finding the x-intercept

To find the x-intercept, we set the value of \( y \) to 0 in the given equation and solve for \( x \). The equation becomes: \[ 3x + 12(0) = -4 \] Simplifying, we find: \[ 3x = -4 \] Dividing by 3, we get the x-intercept: \[ x = -\frac{4}{3} \] So, the x-intercept is \( \left(-\frac{4}{3}, 0\right) \).
02

Finding the y-intercept

To find the y-intercept, we set the value of \( x \) to 0 in the given equation and solve for \( y \). The equation then becomes: \[ 3(0) + 12y = -4 \] Simplifying, we get: \[ 12y = -4 \] Dividing by 12, we find the y-intercept: \[ y = -\frac{1}{3} \] Thus, the y-intercept is \( \left(0, -\frac{1}{3}\right) \).
03

Graphing the Intercepts

With the intercepts found, we can graph the line. Plot the points \( \left(-\frac{4}{3}, 0\right) \) and \( \left(0, -\frac{1}{3}\right) \) on the coordinate plane. Next, draw a line through these intercepts, extending the line across the plane, as it represents the solutions to the equation \( 3x + 12y = -4 \).

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Key Concepts

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

Graphing Linear Equations
Graphing linear equations is all about understanding that a linear equation forms a straight line when plotted on a coordinate plane. To graph the line, you need at least two points. A convenient way to find these points is by using the intercepts. Intercepts are where the line crosses the axes. Once you determine the intercepts, you can plot them and draw a straight line through these points, creating the graph of the equation. This visual representation helps in understanding the relationship between the variables in the equation.
X-Intercept
The x-intercept is the point where the line crosses the x-axis. At this intercept, the value of the y-coordinate is always zero because the point lies on the x-axis itself. To find the x-intercept of a linear equation, simply set the y-variable to zero in the equation and solve for x. For instance, in the equation \( 3x + 12y = -4 \), replacing \( y \) with zero gives \( 3x = -4 \). Solving this, we get \( x = -\frac{4}{3} \). Thus, the x-intercept is \( (-\frac{4}{3}, 0) \). Identifying the x-intercept helps in graphing as it provides one definite point on the line.
Y-Intercept
The y-intercept occurs where the line crosses the y-axis, and at this point, the x-coordinate is always zero. To find the y-intercept, substitute zero for x in the linear equation and solve for y. In the example \( 3x + 12y = -4 \), substitute \( x = 0 \) to get \( 12y = -4 \). Solving this equation results in \( y = -\frac{1}{3} \). Hence, the y-intercept is \( (0, -\frac{1}{3}) \). This gives another specific point on the graph, crucial for drawing the straight line which represents the equation on the graph.
Coordinate Plane
The coordinate plane is a two-dimensional surface where we can graphically show relationships from an equation like \( 3x + 12y = -4 \). It contains two axes: the horizontal x-axis and the vertical y-axis. Each point on this plane is represented by a pair of numbers, known as coordinates, written in the form \( (x, y) \).
  • The x-coordinate indicates the position of the point along the horizontal axis.
  • The y-coordinate indicates the position along the vertical axis.
When graphing linear equations, we plot points derived from the intercepts on this plane and draw a line through them to represent the equation. Understanding the layout and function of a coordinate plane is essential in graphing and interpreting linear equations.

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