2y 6 graph

2y 6 graph DEFAULT

Draw the graph of

Given equation,


x – 2y = 6


⇒–2y = 6 – x




For point, P (2, – 2), x = 2 and y = – 2





= – 2 = LHS


Since, RHS = LHS, therefore, (2, – 2) satisfies x – 2y = 6


For point, Q (4, – 1), x = 4 and y = – 1





= – 1 = LHS


Since, RHS = LHS, therefore, (4, – 1) satisfies x – 2y = 6


For point, Q ( – 2, – 4), x = – 2 and y = – 4





= – 4 = LHS


Since, RHS = LHS, therefore, ( – 2, – 4) satisfies x – 2y = 6


On plotting, P (2, – 2), Q (4, – 1), and R ( – 2, – 4) we get the following graph,



The blue line indicates the required graph of x – 2y = 6


It can be clearly seen from the graph, that the points P (2, – 2), Q (4, – 1), and R ( – 2, – 4) lies on the straight line


Sours: https://goprep.co/q40-draw-the-graph-of-the-equation-x-2y-6-verify-that-each-i-1njxcu

36 Use the Slope–Intercept Form of an Equation of a Line

Graphs

Learning Objectives

By the end of this section, you will be able to:

  • Recognize the relation between the graph and the slope–intercept form of an equation of a line
  • Identify the slope and y-intercept form of an equation of a line
  • Graph a line using its slope and intercept
  • Choose the most convenient method to graph a line
  • Graph and interpret applications of slope–intercept
  • Use slopes to identify parallel lines
  • Use slopes to identify perpendicular lines

Before you get started, take this readiness quiz.

  1. Add: \frac{x}{4}+\frac{1}{4}.
    If you missed this problem, review (Figure).
  2. Find the reciprocal of \frac{3}{7}.
    If you missed this problem, review (Figure).
  3. Solve 2x-3y=12\phantom{\rule{0.2em}{0ex}}\text{for}\phantom{\rule{0.2em}{0ex}}y.
    If you missed this problem, review (Figure).

Recognize the Relation Between the Graph and the Slope–Intercept Form of an Equation of a Line

We have graphed linear equations by plotting points, using intercepts, recognizing horizontal and vertical lines, and using the point–slope method. Once we see how an equation in slope–intercept form and its graph are related, we’ll have one more method we can use to graph lines.

In Graph Linear Equations in Two Variables, we graphed the line of the equation y=\frac{1}{2}x+3 by plotting points. See (Figure). Let’s find the slope of this line.

This figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 8 to 8. The y-axis of the plane runs from negative 8 to 8. The line is labeled with the equation y equals one half x, plus 3. The points (0, 3), (2, 4) and (4, 5) are labeled also. A red vertical line begins at the point (2, 4) and ends one unit above the point. It is labeled “Rise equals 1”. A red horizontal line begins at the end of the vertical line and ends at the point (4, 5). It is labeled “Run equals 2. The red lines create a right triangle with the line y equals one half x, plus 3 as the hypotenuse.

The red lines show us the rise is 1 and the run is 2. Substituting into the slope formula:

\begin{array}{ccc}\hfill m& =\hfill & \frac{\text{rise}}{\text{run}}\hfill \\ \hfill m& =\hfill & \frac{1}{2}\hfill \end{array}

What is the y-intercept of the line? The y-intercept is where the line crosses the y-axis, so y-intercept is \left(0,3\right). The equation of this line is:

The figure shows the equation y equals one half x, plus 3. The fraction one half is colored red and the number 3 is colored blue.

Notice, the line has:

The figure shows the statement “slope m equals one half and y-intercept (0, 3). The slope, one half, is colored red and the number 3 in the y-intercept is colored blue.

When a linear equation is solved for y, the coefficient of the x term is the slope and the constant term is the y-coordinate of the y-intercept. We say that the equation y=\frac{1}{2}x+3 is in slope–intercept form.

The figure shows the statement “m equals one half; y-intercept is (0, 3). The slope, one half, is colored red and the number 3 in the y-intercept is colored blue. Below that statement is the equation y equals one half x, plus 3. The fraction one half is colored red and the number 3 is colored blue. Below the equation is another equation y equals m x, plus b. The variable m is colored red and the variable b is colored blue.

Slope-Intercept Form of an Equation of a Line

The slope–intercept form of an equation of a line with slope m and y-intercept, \left(0,b\right) is,

y=mx+b

Sometimes the slope–intercept form is called the “y-form.”

Use the graph to find the slope and y-intercept of the line, y=2x+1.

Compare these values to the equationy=mx+b.

Solution

To find the slope of the line, we need to choose two points on the line. We’ll use the points \left(0,1\right) and \left(1,3\right).

The slope is the same as the coefficient of x and the y-coordinate of the y-intercept is the same as the constant term.

Use the graph to find the slope and y-intercept of the line y=\frac{2}{3}x-1. Compare these values to the equation y=mx+b.

The figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 8 to 8. The y-axis of the plane runs from negative 8 to 8. The line goes through the points (0, negative 1) and (6, 3).

slope m=\frac{2}{3} and y-intercept \left(0,-1\right)

Use the graph to find the slope and y-intercept of the line y=\frac{1}{2}x+3. Compare these values to the equation y=mx+b.

The figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 8 to 8. The y-axis of the plane runs from negative 8 to 8. The line goes through the points (0, 3) and (negative 6, 0).

slope m=\frac{1}{2} and y-intercept \left(0,3\right)

Identify the Slope and y-Intercept From an Equation of a Line

In Understand Slope of a Line, we graphed a line using the slope and a point. When we are given an equation in slope–intercept form, we can use the y-intercept as the point, and then count out the slope from there. Let’s practice finding the values of the slope and y-intercept from the equation of a line.

Identify the slope and y-intercept of the line with equation y=-3x+5.

Solution

We compare our equation to the slope–intercept form of the equation.

Identify the slope and y-intercept of the line y=\frac{2}{5}x-1.

\frac{2}{5};\left(0,-1\right)

Identify the slope and y-intercept of the line y=-\frac{4}{3}x+1.

-\frac{4}{3};\left(0,1\right)

When an equation of a line is not given in slope–intercept form, our first step will be to solve the equation for y.

Identify the slope and y-intercept of the line with equation x+2y=6.

Solution

This equation is not in slope–intercept form. In order to compare it to the slope–intercept form we must first solve the equation fory.

Identify the slope and y-intercept of the line x+4y=8.

-\frac{1}{4};\left(0,2\right)

Identify the slope and y-intercept of the line 3x+2y=12.

-\frac{3}{2};\left(0,6\right)

Choose the Most Convenient Method to Graph a Line

Now that we have seen several methods we can use to graph lines, how do we know which method to use for a given equation?

While we could plot points, use the slope–intercept form, or find the intercepts for any equation, if we recognize the most convenient way to graph a certain type of equation, our work will be easier. Generally, plotting points is not the most efficient way to graph a line. We saw better methods in sections 4.3, 4.4, and earlier in this section. Let’s look for some patterns to help determine the most convenient method to graph a line.

Here are six equations we graphed in this chapter, and the method we used to graph each of them.

\begin{array}{cccccccc}& & & \mathbf{\text{Equation}}\hfill & & \phantom{\rule{5em}{0ex}}& & \mathbf{\text{Method}}\hfill \\ \text{#1}\hfill & & & x=2\hfill & & \phantom{\rule{5em}{0ex}}& & \text{Vertical line}\hfill \\ \text{#2}\hfill & & & y=4\hfill & & \phantom{\rule{5em}{0ex}}& & \text{Horizontal line}\hfill \\ \text{#3}\hfill & & & \text{−}x+2y=6\hfill & & \phantom{\rule{5em}{0ex}}& & \text{Intercepts}\hfill \\ \text{#4}\hfill & & & 4x-3y=12\hfill & & \phantom{\rule{5em}{0ex}}& & \text{Intercepts}\hfill \\ \text{#5}\hfill & & & y=4x-2\hfill & & \phantom{\rule{5em}{0ex}}& & \text{Slope-intercept}\hfill \\ \text{#6}\hfill & & & y=\text{−}x+4\hfill & & \phantom{\rule{5em}{0ex}}& & \text{Slope-intercept}\hfill \end{array}

Equations #1 and #2 each have just one variable. Remember, in equations of this form the value of that one variable is constant; it does not depend on the value of the other variable. Equations of this form have graphs that are vertical or horizontal lines.

In equations #3 and #4, both x and y are on the same side of the equation. These two equations are of the form Ax+By=C. We substituted y=0 to find the x-intercept and x=0 to find the y-intercept, and then found a third point by choosing another value for x or y.

Equations #5 and #6 are written in slope–intercept form. After identifying the slope and y-intercept from the equation we used them to graph the line.

This leads to the following strategy.

Strategy for Choosing the Most Convenient Method to Graph a Line

Consider the form of the equation.

Determine the most convenient method to graph each line: ⓐ3x+2y=12y=4y=\frac{1}{5}x-4x=-7.

ⓐ intercepts ⓑ horizontal line ⓒ slope–intercept ⓓ vertical line

Determine the most convenient method to graph each line: ⓐx=6y=-\frac{3}{4}x+1y=-84x-3y=-1.

ⓐ vertical line ⓑ slope–intercept ⓒ horizontal line ⓓ intercepts

Graph and Interpret Applications of Slope–Intercept

Many real-world applications are modeled by linear equations. We will take a look at a few applications here so you can see how equations written in slope–intercept form relate to real-world situations.

Usually when a linear equation models a real-world situation, different letters are used for the variables, instead of x and y. The variable names remind us of what quantities are being measured.

The equation F=\frac{9}{5}C+32 is used to convert temperatures, C, on the Celsius scale to temperatures, F, on the Fahrenheit scale.

ⓐ Find the Fahrenheit temperature for a Celsius temperature of 0.
ⓑ Find the Fahrenheit temperature for a Celsius temperature of 20.
ⓒ Interpret the slope and F-intercept of the equation.
ⓓ Graph the equation.

Solution

ⓒ Interpret the slope and F-intercept of the equation.

Even though this equation uses Fand C, it is still in slope–intercept form.

This image shows three lines of equations. The first line reads y equals m x plus b. The second line reads F equals m C plus b and the third line reads F equals nine fifths times C plus 32.

The slope, \frac{9}{5}, means that the temperature Fahrenheit (F) increases 9 degrees when the temperature Celsius (C) increases 5 degrees.

The F-intercept means that when the temperature is 0\text{°} on the Celsius scale, it is 32\text{°} on the Fahrenheit scale.

ⓓ Graph the equation.

We’ll need to use a larger scale than our usual. Start at the F-intercept \left(0,32\right) then count out the rise of 9 and the run of 5 to get a second point. See (Figure).

No Alt Text

The equation h=2s+50 is used to estimate a woman’s height in inches, h, based on her shoe size, s.

ⓐ Estimate the height of a child who wears women’s shoe size 0.
ⓑ Estimate the height of a woman with shoe size 8.
ⓒ Interpret the slope and h-intercept of the equation.
ⓓ Graph the equation.

  1. ⓐ 50 inches
  2. ⓑ 66 inches
  3. ⓒ The slope, 2, means that the height, h, increases by 2 inches when the shoe size, s, increases by 1. The h-intercept means that when the shoe size is 0, the height is 50 inches.

  4. The figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane represents the variable s and runs from negative 2 to 15. The y-axis of the plane represents the variable h and runs from negative 1 to 80. The line begins at the point (0, 50) and goes through the points (8, 66).

The equation T=\frac{1}{4}n+40 is used to estimate the temperature in degrees Fahrenheit, T, based on the number of cricket chirps, n, in one minute.

ⓐ Estimate the temperature when there are no chirps.
ⓑ Estimate the temperature when the number of chirps in one minute is 100.
ⓒ Interpret the slope and T-intercept of the equation.
ⓓ Graph the equation.

  1. ⓐ 40 degrees
  2. ⓑ 65 degrees
  3. ⓒ The slope, \frac{1}{4}, means that the temperature Fahrenheit (F) increases 1 degree when the number of chirps, n, increases by 4. The T-intercept means that when the number of chirps is 0, the temperature is 40\text{°}.

  4. The figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane represents the variable n and runs from 10 to 140 The y-axis of the plane represents the variable T and runs from negative 5 to 75. The line begins at the point (0, 40) and goes through the point (100, 65).

The cost of running some types business has two components—a fixed cost and a variable cost. The fixed cost is always the same regardless of how many units are produced. This is the cost of rent, insurance, equipment, advertising, and other items that must be paid regularly. The variable cost depends on the number of units produced. It is for the material and labor needed to produce each item.

Stella has a home business selling gourmet pizzas. The equation C=4p+25 models the relation between her weekly cost, C, in dollars and the number of pizzas, p, that she sells.

ⓐ Find Stella’s cost for a week when she sells no pizzas.
ⓑ Find the cost for a week when she sells 15 pizzas.
ⓒ Interpret the slope and C-intercept of the equation.
ⓓ Graph the equation.

Sam drives a delivery van. The equation C=0.5m+60 models the relation between his weekly cost, C, in dollars and the number of miles, m, that he drives.

ⓐ Find Sam’s cost for a week when he drives 0 miles.
ⓑ Find the cost for a week when he drives 250 miles.
ⓒ Interpret the slope and C-intercept of the equation.
ⓓ Graph the equation.

  1. ⓐ ?60
  2. ⓑ ?185
  3. ⓒ The slope, 0.5, means that the weekly cost, C, increases by ?0.50 when the number of miles driven, n, increases by 1. The C-intercept means that when the number of miles driven is 0, the weekly cost is ?60

  4. The figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane represents the variable m and runs from negative 10 to 400. The y-axis of the plane represents the variable C and runs from negative 10 to 300. The line begins at the point (0, 65) and goes through the point (250, 185).

Loreen has a calligraphy business. The equation C=1.8n+35 models the relation between her weekly cost, C, in dollars and the number of wedding invitations, n, that she writes.

ⓐ Find Loreen’s cost for a week when she writes no invitations.
ⓑ Find the cost for a week when she writes 75 invitations.
ⓒ Interpret the slope and C-intercept of the equation.
ⓓ Graph the equation.

  1. ⓐ ?35
  2. ⓑ ?170
  3. ⓒ The slope, 1.8, means that the weekly cost, C, increases by ?1.80 when the number of invitations, n, increases by 1.80.
    The C-intercept means that when the number of invitations is 0, the weekly cost is ?35.;

  4. The figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane represents the variable n and runs from negative 10 to 400. The y-axis of the plane represents the variable C and runs from negative 10 to 300. The line begins at the point (0, 35) and goes through the point (75, 170).

Use Slopes to Identify Parallel Lines

The slope of a line indicates how steep the line is and whether it rises or falls as we read it from left to right. Two lines that have the same slope are called parallel lines. Parallel lines never intersect.

The figure shows three pairs of lines side-by-side. The pair of lines on the left run diagonally rising from left to right. The pair run side-by-side, not crossing. The pair of lines in the middle run diagonally dropping from left to right. The pair run side-by-side, not crossing. The pair of lines on the right run diagonally also dropping from left to right, but with a lesser slope. The pair run side-by-side, not crossing.

We say this more formally in terms of the rectangular coordinate system. Two lines that have the same slope and different y-intercepts are called parallel lines. See (Figure).

Verify that both lines have the same slope, m=\frac{2}{5}, and different y-intercepts.

The figure shows two lines graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 8 to 8. The y-axis of the plane runs from negative 8 to 8. One line goes through the points (negative 5,1) and (5,5). The other line goes through the points (negative 5, negative 4) and (5,0).

What about vertical lines? The slope of a vertical line is undefined, so vertical lines don’t fit in the definition above. We say that vertical lines that have different x-intercepts are parallel. See (Figure).

Vertical lines with diferent x-intercepts are parallel.

The figure shows two vertical lines graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 8 to 8. The y-axis of the plane runs from negative 8 to 8. One line goes through the points (2,1) and (2,5). The other line goes through the points (5, negative 4) and (5,0).

Parallel Lines

Parallel lines are lines in the same plane that do not intersect.

  • Parallel lines have the same slope and different y-intercepts.
  • If {m}_{1} and {m}_{2} are the slopes of two parallel lines then{m}_{1}={m}_{2}.
  • Parallel vertical lines have different x-intercepts.

Let’s graph the equations y=-2x+3 and 2x+y=-1 on the same grid. The first equation is already in slope–intercept form: y=-2x+3. We solve the second equation for y:

\begin{array}{ccc}\hfill 2x+y& =\hfill & -1\hfill \\ \hfill y& =\hfill & -2x-1\hfill \end{array}

Graph the lines.

The figure shows two lines graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 8 to 8. The y-axis of the plane runs from negative 8 to 8. One line goes through the points (negative 4, 7) and (3, negative 7). The other line goes through the points (negative 2, 7) and (5, negative 7).

Notice the lines look parallel. What is the slope of each line? What is the y-intercept of each line?

\begin{array}{cccccccccc}\hfill y& =\hfill & mx+b\hfill & & \phantom{\rule{5em}{0ex}}& & & \hfill y& =\hfill & mx+b\hfill \\ \hfill y& =\hfill & -2x+3\hfill & & \phantom{\rule{5em}{0ex}}& & & \hfill y& =\hfill & -2x-1\hfill \\ \hfill m& =\hfill & -2\hfill & & \phantom{\rule{5em}{0ex}}& & & \hfill m& =\hfill & -2\hfill \\ \hfill b& =\hfill & 3,\text{(0, 3)}\hfill & & \phantom{\rule{5em}{0ex}}& & & \hfill b& =\hfill & -1,\text{(0, −1)}\hfill \end{array}

The slopes of the lines are the same and the y-intercept of each line is different. So we know these lines are parallel.

Since parallel lines have the same slope and different y-intercepts, we can now just look at the slope–intercept form of the equations of lines and decide if the lines are parallel.

Use slopes and y-intercepts to determine if the lines 3x-2y=6 and y=\frac{3}{2}x+1 are parallel.

Sours: https://opentextbc.ca/elementaryalgebraopenstax/chapter/use-the-slope-intercept-form-of-an-equation-of-a-line/
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Properties of a straight line

Rearrange:

Rearrange the equation by subtracting what is to the right of the equal sign from both sides of the equation :

                     6*x-2*y-(6)=0 

Step  1  :

Pulling out like terms :

 1.1     Pull out like factors :

   6x - 2y - 6  =   2 • (3x - y - 3) 

Equation at the end of step  1  :

Step  2  :

Equations which are never true :

 2.1      Solve :    2   =  0

This equation has no solution.
A a non-zero constant never equals zero.

Equation of a Straight Line

 2.2     Solve   3x-y-3  = 0

Tiger recognizes that we have here an equation of a straight line. Such an equation is usually written y=mx+b ("y=mx+c" in the UK).

"y=mx+b" is the formula of a straight line drawn on Cartesian coordinate system in which "y" is the vertical axis and "x" the horizontal axis.

In this formula :

y tells us how far up the line goes
x tells us how far along
m is the Slope or Gradient i.e. how steep the line is
b is the Y-intercept i.e. where the line crosses the Y axis

The X and Y intercepts and the Slope are called the line properties. We shall now graph the line  3x-y-3  = 0 and calculate its properties

Graph of a Straight Line :

Calculate the Y-Intercept :

Notice that when x = 0 the value of y is 3/-1 so this line "cuts" the y axis at y=-3.00000

  y-intercept = 3/-1 = -3.00000

Calculate the X-Intercept :

When y = 0 the value of x is 1/1 Our line therefore "cuts" the x axis at x= 1.00000

  x-intercept = 3/3 = 1

Calculate the Slope :

Slope is defined as the change in y divided by the change in x. We note that for x=0, the value of y is -3.000 and for x=2.000, the value of y is 3.000. So, for a change of 2.000 in x (The change in x is sometimes referred to as "RUN") we get a change of 3.000 - (-3.000) = 6.000 in y. (The change in y is sometimes referred to as "RISE" and the Slope is m = RISE / RUN)

  Slope = 3

Geometric figure: Straight Line

  1.   Slope = 3
  2.   x-intercept = 3/3 = 1
  3.   y-intercept = 3/-1 = -3.00000
Sours: https://www.tiger-algebra.com/drill/6x-2y=6/
Find four different solutions of the linear equation x+2y=6?
Hint: In this question, we are given an equation of a line and we have been asked to draw the line on a graph. In order to draw it on a graph, find the point that lies on the graph. You can find the points by keeping different values of $x$ and then using them, find the values of $y$. Find at least three points and plot them on a graph paper. Join the points using a scale and you will have your required line.

Complete step-by-step solution:
In this question, we are given an equation of a line $x + 2y = 6$ and we have been asked to plot this on the graph.
Let us find the points.
1) Let $x = 0$. Put this in the equation.
$ \Rightarrow 0 + 2y = 6$
Shifting to find the value of $y$,
$ \Rightarrow y = \dfrac{6}{2} = 3$
It gives us the point $\left( {0,3} \right)$. We will name it A.
Therefore, our first point is A$\left( {0,3} \right)$.
2) Let $y = 0$. Put this in the equation.
$ \Rightarrow x + 0 = 6$
$ \Rightarrow x = 6$
It gives us the point $\left( {6,0} \right)$. We will name it as B.
Therefore, our first point is B$\left( {6,0} \right)$.
3) Let $x = 2$. Put this in the equation.
$ \Rightarrow 2 + 2y = 6$
Shifting to find the value of $y$,
$ \Rightarrow 2y = 6 - 2 = 4$
$ \Rightarrow y = \dfrac{4}{2} = 2$
It gives us the point $\left( {2,2} \right)$. We will name it as C.
Therefore, our first point is C$\left( {2,2} \right)$.
Let us plot the three points on the graph.

Next step is to connect these points. Keep in mind to use a scale.

Hence, this is our final answer.

Note: In such questions, always find at least three points as there is step marking in exams and they need you to find three points.
An easy trick in such questions is to find points on the two axes by keeping $x = 0$ and then, $y = 0$. Only by plotting these two points, you will get a perfect line.
Sours: https://www.vedantu.com/question-answer/graph-x-+-2y-6-by-plotting-points-class-8-maths-cbse-5feab15141231c3a7848fa0d

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Sours: https://www.symbolab.com/solver/functions-graphing-calculator/x-2y%3D-6
Draw the graph of the linear equation `x=2y-6`.

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Now discussing:

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