GRAPHING LINEAR EQUATIONS IN SLOPE INTERCEPT FORM

If a linear equation is in the form

y  =  mx + b,

then it is called slope-intercept form. 

Here, 'm' stands for slope and 'b' stands for y-intercept. 

We already know that the graph any linear equation will be a straight line.

When we have a linear equation in slope-intercept form, we can sketch the graph (straight line) of the equation using the slope 'm' and y-intercept 'b'. 

y-intercept :

y-intercept is nothing but the value at where the line intersects y-axis. 

Slope :

Slope is sometimes referred to as "rise over run".

That is, 

slope  =  rise/run

Because the fraction consists of the rise (the change in y, going up or down) divided by the run (the change in x, going from left to the right).

Rising / Falling / Horizontal / Vertical Line :

(i) If the slope of a line is positive, then the line will be going (from left to right) up and it is called rising line.

(ii) If the slope of a line is positive, then the line will be going (from left to right) down and it is called falling line. 

(iii) If the slope is zero, the line will be horizontal.

(iv) If the slope is undefined, the line will be vertical. 

Example 1 :

Graph the following linear equation.

y  =  2x + 1

Solution :

The equation 'y  =  2x + 1' is in the form of 

y  =  mx + b

Then, 

m  =  2

b  =  1

Because slope '2' is a positive value, the line will be a rising line.

And also,

rise/run  =  2

rise/run  =  2/1

Then,

rise  =  2

run  =  1

Because the y-intercept is 1, the line will intersect y-axis at 1. 

Graphing : 

Step 1 :

Plot the y-intercept at (0, 1). 

Step 2 :

Because the run is 1, move 1 unit to the right from (0, 1).

Step 3 :

Because the rise is 2 and the line is rising line, move 2 units up from the position reached in step 2.   

Now, you are at (1, 3). 

Connect the points (0, 1) and (1, 3) to get the line.  

Example 2 :

Graph the following linear equation.

y  =  -1.5x + 2

Solution :

The equation 'y  =  -1.5x + 2' is in the form of 

y  =  mx + b

Then, 

m  =  -1.5

b  =  2

Because slope '-1.5' is a negative value, the line will be a falling line.

And also,

rise/run  =  1.5

rise/run  =  3/2 

Then,

rise  =  3

run  =  2

Because the y-intercept is 2, the line will intersect y-axis at 2. 

Graphing : 

Step 1 :

Plot the y-intercept at (0, 2). 

Step 2 :

Because the run is 2, move 2 units to the right from (0, 2).

Step 3 :

Because the rise is 3 and the line is falling line, move 3 units down from the position reached in step 2.   

Now, you are at (2, -1). 

Connect the points (0, 2) and (2, -1) to get the line.  

Example 3 :

Graph the following linear equation.

x - y - 2  =  0

Solution :

The equation 'x - y - 2  =  0' is not slope -intercept form. 

Write the given equation in slope-intercept form. 

x - y - 2  =  0

Add y to each side. 

x - 2  =  y

or 

y  =  x - 2

The equation 'y  =  x - 2' is in the form of 

y  =  mx + b

Then, 

m  =  1

b  =  -2

Because slope '1' is a positive value, the line will be a rising line.

And also,

rise/run  =  1

rise/run  =  1/1 

Then,

rise  =  1

run  =  1

Because the y-intercept is -2, the line will intersect y-axis at -2. 

Graphing : 

Step 1 :

Plot the y-intercept at (0, -2). 

Step 2 :

Because the run is 1, move 1 unit to the right from (0, -2).

Step 3 :

Because the rise is 1 and the line is rising line, move 1 unit up from the position reached in step 2.   

Now, you are at (1, -1). 

Connect the points (0, -2) and (1, -1) to get the line.  

Example 4 :

Graph the following linear equation.

5x + y  =  3

Solution :

The equation '5x + y  =  3' is not slope -intercept form. 

Write the given equation in slope-intercept form. 

5x + y  =  3

Subtract 5x from each side. 

y  =  3 - 5x

y  =  -5x + 3

The equation 'y  =  -5x + 3' is in the form of 

y  =  mx + b

Then, 

m  =  -5

b  =  3

Because slope '-5' is a negative value, the line will be a falling line.

And also,

rise/run  =  5

rise/run  =  5/1 

Then,

rise  =  5

run  =  1

Because the y-intercept is 3, the line will intersect y-axis at 3. 

Graphing : 

Step 1 :

Plot the y-intercept at (0, 3). 

Step 2 :

Because the run is 1, move 1 unit to the right from (0, 3).

Step 3 :

Because the rise is 5 and the line is falling line, move 5 units down from the position reached in step 2.   

Now, you are at (1, -2). 

Connect the points (0, 3) and (1, -2) to get the line.  

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