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Chapter 4: Problem 99

In the following exercises, graph by plotting points. \(x-6 y=3\)

Short Answer

Expert verified
Rewrite in slope-intercept form \(y = \frac{1}{6}x - \frac{1}{2}\), choose \(x = 0\) and \(x = 6\), plot points and draw the line.

Step by step solution

01

Rewrite the Equation in Slope-Intercept Form

The original equation is given as \(x - 6y = 3\). To rewrite this in slope-intercept form \(y = mx + b\), solve for \(y\): 1. Subtract \(x\) from both sides: \(-6y = -x + 3\). 2. Divide by \(-6\) to isolate \(y\): \(y = \frac{1}{6}x - \frac{1}{2}\).
02

Choose Points for the Graph

To plot the line, choose at least two values for \(x\) to find corresponding \(y\) values. For simplicity, select \(x = 0\) and \(x = 6\): 1. When \(x = 0\), \(y = \frac{1}{6}(0) - \frac{1}{2} = -\frac{1}{2}\). 2. When \(x = 6\), \(y = \frac{1}{6}(6) - \frac{1}{2} = 1 - \frac{1}{2} = \frac{1}{2}\).
03

Plot the Points

Plot the obtained points on a graph: 1. Point A: \((0, -\frac{1}{2})\). 2. Point B: \((6, \frac{1}{2})\).
04

Draw the Line

Draw a straight line passing through both points \((0, -\frac{1}{2})\) and \((6, \frac{1}{2})\). This is the graph of the equation \(x - 6y = 3\). The line represents all the solutions to the equation.

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Key Concepts

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

slope-intercept form
The slope-intercept form is an essential way to write linear equations. It's written as: ewlineewlineewlineewline ewline ewlineewline ewline ewline ewline y = mx + b, where:
  • y is the output or dependent variable,
  • x is the input or independent variable,
  • m is the slope of the line,
  • b is the y-intercept (the point where the line crosses the y-axis).
To convert an equation to slope-intercept form, solve for y. In the example with the equation ewlineewline ewline ewline ewline x - 6y = 3, you need to:
  1. Subtract x from both sides: ewlineewline-6y = -x + 3.
  2. Divide every term by -6: ewlineewline y = (1/6)x - 1/2.
Now, it's in slope-intercept form where the slope (m) is 1/6 and the y-intercept (b) is -1/2.
plotting points
Plotting points is a crucial step for graphing linear equations. To plot points, you need pairs of x and y values. Let's use the equation y = (1/6)x - 1/2. Choose simple x values to find corresponding y values.
  • When x = 0, solve for y: ewlineewline y = (1/6)(0) - 1/2 = -1/2.
  • When x = 6, solve for y: ewlineewline y = (1/6)(6) - 1/2 = 1/2.
Two points are now ewlineewline(0, -1/2) and ewline6, 1/2). Plot these points on an xy-plane. First, find (0, -1/2) on the vertical y-axis. Then, find (6, 1/2) by moving six units right and 1/2 unit up. This makes it easy to visualize the line.
solving for y
Solving for y is essential when converting an equation to slope-intercept form. Start with the original form x - 6y = 3 and isolate y in the following steps:
  1. Move the x term to the right side by subtracting x from both sides: ewline ewlineewline-6y = -x + 3.
  2. Divide by -6 to solve for y: ewliney = (1/6)x - 1/2.
Now, y is isolated and the equation is in y = mx + b slope-intercept form. This makes it easy to graph the equation and understand the line鈥檚 characteristics, like its slope and y-intercept.
linear graph
A linear graph visually represents solutions to linear equations. To graph the line for y = (1/6)x - 1/2:
  • Plot at least two points, such as (0, -1/2) and (6, 1/2).
  • Use a ruler to draw a straight line through the points.
This line includes all solutions to y = (1/6)x - 1/2. Every point on the line is an ewlineewline x,y pair that satisfies the equation. Linear graphs show the direct relationship between x and y, making it easier to understand changes in values.

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Most popular questions from this chapter

Determine whether each ordered pair is a solution to the inequality \(x+y>4\) : (a) (5,1) (b) (-2,6) c) (3,2) (d) (10,-5) (e) (0,0)

Find the equation of each line. Write the equation in slope-intercept form. Containing the points (-5,-3) and (4,-6)

Graph the linear inequality \(x>-2\)

Find an equation of a line perpendicular to the given line and contains the given point. Write the equation in slope-intercept form. line \(y=\frac{3}{4} x-2,\) point(-3,4)

Find an equation of a line perpendicular to the given line and contains the given point. Write the equation in slope-intercept form. line \(2 x+5 y=6,\) point(0,0)
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