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Chapter 8: Problem 28

For Problems 1-36, graph each linear equation. (Objective 2) $$ 3 x+5 y=-9 $$

Short Answer

Expert verified
Graph the line by plotting points at \((0, -\frac{9}{5})\) and \((5, -\frac{24}{5})\) and drawing a line through them.

Step by step solution

01

Write the Equation in Slope-Intercept Form

To graph the equation, first rewrite it in slope-intercept form, which is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. Start by isolating \( y \) in the equation \( 3x + 5y = -9 \). Subtract \( 3x \) from both sides to get \( 5y = -3x - 9 \). Then, divide every term by 5 to solve for \( y \): \[ y = -\frac{3}{5}x - \frac{9}{5} \].
02

Identify the Slope and Y-intercept

From the equation \( y = -\frac{3}{5}x - \frac{9}{5} \), identify the slope \( m \) as \(-\frac{3}{5}\) and the y-intercept \( b \) as \(-\frac{9}{5}\). The y-intercept is the point \( (0, -\frac{9}{5}) \) on the graph.
03

Plot the Y-intercept

On the Cartesian plane, plot the y-intercept \( (0, -\frac{9}{5}) \). This is the point where the line crosses the y-axis.
04

Use the Slope to Plot Another Point

Starting from the y-intercept, use the slope \( -\frac{3}{5} \) to find another point. The slope indicates that for every 3 units moved down (because it is negative), move 5 units to the right. From \( (0, -\frac{9}{5}) \), moving 5 units right and 3 units down leads you to the point \( (5, -\frac{24}{5}) \). Plot this point.
05

Draw the Line

With the two points \( (0, -\frac{9}{5}) \) and \( (5, -\frac{24}{5}) \) plotted, draw a straight line through them. Ensure that the line extends in both directions and has arrows to show it continues beyond these points.

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Key Concepts

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

Slope-Intercept Form
When dealing with linear equations, the slope-intercept form is a common way to express them. This form makes it easy to graph the equation and understand the line's properties. The general formula for the slope-intercept form is \( y = mx + b \), where:
  • \( m \) represents the slope of the line, which indicates how steep the line is.
  • \( b \) is the y-intercept, the point where the line crosses the y-axis.
To rewrite an equation in this form, you need to solve for \( y \). For example, with the equation \( 3x + 5y = -9 \), isolate \( y \) to get \( y = -\frac{3}{5}x - \frac{9}{5} \). Now, you have a clear picture of the line's slope and its y-intercept, making it easier to graph.
Cartesian Plane
The Cartesian plane is a two-dimensional surface used for graphing equations. It consists of two perpendicular axes:
  • The horizontal axis, called the x-axis.
  • The vertical axis, called the y-axis.
These axes divide the plane into four quadrants, each containing positive and negative values for x and y coordinates. Origin \( (0,0) \) is where both axes intersect. Understanding the Cartesian plane allows you to visually represent equations and comprehend how changes in formulas alter their graphs. In our example, using the plane helps in locating points from the linear equation by starting at the y-intercept and using the slope to find other points.
Plotting Points
Plotting points on the Cartesian plane is a crucial step in graphing equations. Each point is defined by a pair of coordinates \( (x, y) \), indicating its location. To plot a point, simply:
  • Find the x-coordinate on the x-axis and mark it.
  • From that mark, move vertically to locate the y-coordinate on the y-axis.
  • Mark this intersection as a point on the graph.
Once you plot the y-intercept from the slope-intercept form, use the slope to determine and plot another point. For example, with a slope of \(-\frac{3}{5}\), you go 3 units down for every 5 units right starting from the y-intercept. Plot these points to guide you in drawing the line, illustrating the linear relationship presented by the equation.

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Most popular questions from this chapter

Find the equation of the line that contains the two given points. Express equations in the form \(A x+B y=C\), where \(A, B\), and \(C\) are integers. (Objective \(1 \mathrm{~b}\) ) \((1,4)\) and \((9,10)\)

For Problems \(33-44\), determine the slope and \(y\) intercept of the line represented by the given equation, and graph the line. $$ 9 x+7 y=0 $$

For Problems \(23-32\), find the equation of the line with the given slope and \(y\) intercept. Leave your answers in slope-intercept form. $$ m=\frac{3}{5} \text { and } b=2 $$

For Problems \(33-44\), determine the slope and \(y\) intercept of the line represented by the given equation, and graph the line. $$ y=-2 x-5 $$

For Problems \(23-32\), find the equation of the line with the given slope and \(y\) intercept. Leave your answers in slope-intercept form. $$ m=-1 \text { and } b=\frac{5}{2} $$
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