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Construct a graph corresponding to the linear equation y=2x−6.

In an algebraic equation, the linear equation has the highest degree of 1, thus the reason it is named linear equation. A linear equation can be represented in a 1 variable and 2 variable form. Graphically, a linear equation is demonstrated by a straight line on the x-y coordinate system.

A linear equation comprises two elements i.e. constants and variables. In one variable, the standard linear equation is represented as:

\[ax+b=0, \ where \ a ≠ 0 \ and \ x \ is \ the \ variable.\]

With two variables, the standard linear equation is represented as:

\[ax+by+c=0, \ where \ a ≠ 0, \ b ≠ 0 \ and \ x \ and \ y \ are \ the \ variable.\]

In this question, we have to draw the graph for the given linear equation by putting the values of $x$ to get the $y$ coordinates.

In the linear form of an equation, we can easily find both x-intercept and y-intercept, especially when dealing with systems of two linear equations. Following is the example of a linear equation in $2$ variables:

\[ 4x+8y=2 \]

Expert Answer

To plot the graph of the given equation in question, we have to find the respective $x$ and $y$ coordinates by putting different values of $x$ to get the value of $y$.

For this, we have the equation:

\[ y=2x-6 \]

First putting the value of $x=-3$, we get:

\[ y=2 \left (-3 \right)- 6\]

\[ y=-6- 6 \]

\[ y=-12 \]

We get the coordinates $(-3,-12)$.

Now putting the value of $x=-2$, we get:

\[ y=2 \left (-2\right)- 6\]

\[ y=-4-6 \]

\[ y=-10 \]

We get the coordinates $(-2,-10)$.

Putting the value of $x=-1$, we get:

\[ y=2 \left (-1\right)- 6 \]

\[ y=-2-6 \]

\[ y=-8 \]

We get the coordinates $(-1,-8)$.

Putting the value of $x=0$, we get:

\[ y=2\left (0\right)- 6 \]

\[ y=0- 6 \]

\[ y=-6 \]

We get the coordinates $(0,-6)$.

When $x=1$:

\[ y=2\left (1\right)- 6 \]

\[ y=2-6 \]

\[ y=-4 \]

We get the coordinates $(1,-4)$.

When $x=2$:

\[y=2\left(2\right)- 6\]

\[y=4- 6\]

\[y=-2\]

We get the coordinates $(2,-2)$.

When $x=3$:

\[y=2\left(3\right)- 6\]

\[y=6- 6\]

\[y=0\]

We get the coordinates $(3,0)$.

So our required coordinates are:

\[ (-3,-12),(-2,-10),(-1,-8), (0,-6),(1,-4), (2,-2),(3,0) \]

Now plotting these coordinates on the graph, we get the following graph:

linear equations graph

Figure 1

Numerical Results

The required coordinates for plotting the graph of equation $y=2x-6$ are $ (-3,-12),(-2,-10),(-1,-8) ,(0,-6),(1,-4),(2,-2), (3,0)$, as shown in the following graph:

linear equations graph

Figure 2

Example

Plot the graph for the equation $y=2x+1$

Solution: First we will find its respective y-coordinates by putting values of $x$:

when $x=-1$

\[y=2(-1)+1=-1\]

when $x=0$

\[y=2(0)+1=1\]

when $x=1$

\[y=2(1)+1=-3\]

when $x=2$

\[y=2(2)+1=5\]

So our required coordinates are $(-1,-1), (0,1), (1,3), (2,5)$. Now, plotting these coordinates on a graph, we get the following graph:

linear equation

Figure 3

Image/Mathematical drawings are created in Geogebra.

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