Linear Equations
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Sketching Linear Graphs


How do I graph a linear relationship?

To graph any liner relationship you only need two points that are on the line.  You can use any two points from a table of values, or substitute in any two values of $x$x into the equation and solve for corresponding $y$y-value to create your own two points.  Often, using the intercepts is one of the easiest ways to sketch the line.

Example - sketch from table of values

x 1 2 3 4
y 3 5 7 9

To sketch from a table of values, we need just any two points from the table.  From this table we have 4 coordinates, $\left(1,3\right)$(1,3), $\left(2,5\right)$(2,5), $\left(3,7\right)$(3,7), $\left(4,9\right)$(4,9).  

Drag the $2$2 of the points on this interactive to the correct positions and graph this linear relationship.  

Example - sketch from any two points

If we are given the equation of a linear relationship, like $y=3x+5$y=3x+5, then to sketch it we need two points.  We can pick any two points we like.  

Start by picking any two $x$x-values you like, often the $x$x-value of $0$0 is a good one to pick because the calculation for y can be quite simple.  For our example, $y=3x+5$y=3x+5 becomes $y=0+5$y=0+5, $y=5$y=5.  This gives us the point $\left(0,5\right)$(0,5)

Similarly look for other easy values to calculate such as $1$1, $10$10, $2$2.  I'll pick $x=1$x=1.  Then for $y=3x+5$y=3x+5, we have $y=3\times1+5$y=3×1+5, $y=8$y=8.This gives us the point $\left(1,8\right)$(1,8)

Now we plot the two points and create a line.  


Example - sketch from the intercepts

The general form of a line is great for identifying both the x and y intercepts easily. 

For example, the line $3y+2x-6=0$3y+2x6=0

The x intercept happens when the $y$y value is $0$0$3y+2x-6=0$3y+2x6=0 $0+2x-6=0$0+2x6=0 $2x=6$2x=6 $x=3$x=3 The y intercept happens when the $x$x value is $0$0$3y+2x-6=0$3y+2x6=0 $3y+0-6=0$3y+06=0 $3y=6$3y=6 $y=2$y=2

From here it is pretty easy to sketch, we find the $x$x intercept $3$3, and the $y$y intercept $2$2, and draw the line through both.

Example - sketch from the gradient and a point

Start by plotting the single point that you are given.  

Remembering that gradient is a measure of change in the rise per change in run, we can step out one measure of the gradient from the original point given. 

For a gradient of $4$4  $1$1 unit across and $4$4 units up. For a gradient of $-3$3 $1$1 unit across and $3$3 units down. For a gradient of $\frac{1}{2}$12 $1$1 unit across and $\frac{1}{2}$12 unit up.  

The point can be any point $\left(x,y\right)$(x,y), or it could be an intercept.  Either way, plot the point, step out the gradient and draw your line!

For example, plot the line with gradient $-2$2 and has $y$y intercept of $4$4.  

Start with the point, ($y$y intercept of $4$4) Step out the gradient, (-$2$2 means $2$2 units down)

Draw the line

To sketch linear graphs, it's easiest to substitute in values to find coordinates to put it in gradient-intercept form. 

The Gradient-Intercept Form


where $m$m is the gradient and $b$b is the $y$y-intercept

Our graphs may not always be in this form so we may need to rearrange the equation to make $y$y the subject (that means $y$y is on one side of the equation and everything else is on the other side).


What happens if we're not given the equation of a line?

Sometimes, it doesn't matter. We can sketch a straight line on a graph just by knowing a couple of its features such as a point that lies on the line and it's gradient. At other times, we may need to generate an equation before we sketch it. So other than the gradient-intercept form, we can use:

  • Gradient-point formula:  $y-y_1=m\left(x-x_1\right)$yy1=m(xx1)


  • Two point formula: $\frac{y-y_1}{x-x_1}=\frac{y_2-y_1}{x_2-x_1}$yy1xx1=y2y1x2x1


Ok let's look at this in action with some examples.



Question 1

Plot the graph of the line whose gradient is $-3$3 and passes through the point $\left(-2,4\right)$(2,4).

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Question 2

Graph the linear equation $y=3x-1$y=3x1 using the point $Y$Y as the $y$y-intercept.

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Question 3

Graph the linear equation $-6x+3y+24=0$6x+3y+24=0 by finding any two points on the line.

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Horizontal or Vertical Graphs

On horizontal lines, the $y$y value is always the same for every point on the line.

On vertical lines, the $x$x value is always the same for every point on the line.


Question 4

Graph the line $y=-3$y=3.

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