Graphic
Solution
of
Linear Equations

Two linear (line) equations, when graphed on a coordinate system can be:

Parallel
Coinciding
Intersecting
No points of intersection
All points of intersection
One point of intersection

 

The solution that we obtain by graphing these equations will be the SAME as the one that was obtained by the Algebraic method. By comparing the two methods, you should come to understand the graph as a "picture representation" of its algebraic solution.
 
The equations are:
10x + 20y =
120
12x + 4y =
44

 

Follow these directions for both equations above.

The four steps to a graphic solution using the slope-intercept method are:

1. Solve for y in terms of x.

2. Identify the y-intercept and graph it.

3. Identify the slope and plot it on the graph.

4. Draw the line that connects all points.

 

We will begin with the equation:
10x
+
20y
=
120
1. Solve for y:
10x
+
20y
=
120
-10x
      
-10x
   
20y
=
(120 - 10x)
   
20y/20
=
(120 - 10x)/20
   
y
=
6 - 1/2 x
   
y
=
- 1/2 x + 6

 

2. Finding the y-intercept: This is the point where the graph crosses the y - axis. The general equation for the slope-intercept form is y = mx + b where "b" represents the y-interecept and "m" represents the slope of the line. If you take our equation and place it under the general form, the y-intercept would be +6. (Always include the sign to the left of the number).
y
=
m x
+
b
y
=
- 1/2 x
+
6
Now we know that on the y-axis, the graph of this line will cross at +6. To graph this point, start at the origin (the center of the two axis) and count up (because it is positive) 6 units on the y-axis. Place at point at that location.

3. Next, we will identify the slope of the equation. Again we will use the general form of slope - intercept y = mx + b where "m" represents the slope.
y
=
m x
+
b
y
=
- 1/2 x
+
6
To locate the slope, look directly under the "m" above and you will see the fraction of -1/2. The negative sign can be placed in the numerator or denominator. We will place it in the numerator this time. Now we will plot the slope of -1/+2 using its definition of:

 

 

The slope for -1/+2 would therefore be "down 1(for the negative change in y)"and "right 2 (for the positive change in x)"
Always begin to plot the slope at the y - intercept (which is where you placed the point in step 2 above). The numerator of the fraction is -1 so we will go down one on the y - axis. The denominator of the fraction is +2 so we will go right 2 in the direction of the x - axis.

4. Now connect all the points that you found in steps 2 and 3 above. Label the graph with the original equation and it is complete.

Let's Begin the Second Linear Equation
The equation is:
12x + 4y = 44
The work for this problem should be placed on the SAME graph as the one that we did above. However, to make it easier to understand the steps, I will do this problem on its own graph and them put both graphs together for you at the end.

1. Solve for y:
12x
+
4y
=
44
-12x
      
-12x
   
4y
=
(44 - 12x)
   
4y/4
=
(44 - 12x)/4
   
y
=
11 - 3x
   
y
=
- 3x + 11

 

2. Finding the y-intercept: This is the point where the graph crosses the y - axis. The general equation for the slope - intercept is y = mx + b where "b" represents the y-interecept and "m" represents the slope of the line. If you take our equation and place it under the general form, the y-intercept would be +11. (Always include the sign to the left of the number).
y
=
m x
+
b
y
=
- 3 x
+
11
Now we know that on the y-axis, the graph of this line will cross at +11. To graph this point, start at the origin (the center of the two axis) and count up (because it is positive) 11 units on the y-axis. Place at point at that location.

3. Next, we will identify the slope of the equation. Again we will use the general form of slope - intercept y = mx + b where "m" represents the slope.
y
=
m x
+
b
y
=
- 3 x
+
11
To locate the slope, look directly under the "m" above and you will see - 3. Since slope must always contain two numbers, we will write it with a denominator of 1 (which will not change the value of - 3). The negative sign can be placed in the numerator or the denominator. We will place it in the numerator this time. Now we will plot the slope of -3/+1 using its definition of:

The slope of -3/+1 would be "down 3 (for the negative change in y)" and "right 1 (for the positive change in x)"

 

Always begin to plot the slope at the y - intercept (which is where you placed the point in step 2 above). The numerator of the fraction is -3 so we will go down three on the y - axis. The denominator of the fraction is +1 so we will go right 1 in the direction of the x - axis.

4. Now connect all the points that you found in steps 2 and 3 above. Label the graph with the original equation and it is complete.

The Last Step
Now that we have finished graphing both linear equations, we will place them on the SAME graph so that we can find their point of intersection. Look at the point that both graphs cross at. To determine what the coordinates (the solution) are, start at the origin (where the x and y axis meet) and find the x and y value of that point of intersection. The x value is written first and the y value is written second. Therefore, the solution would be (2,5).

 

Rollover demonstration

 
Animation of both graphs
How does this answer compare with the one competed in the algebraic solution? Check it out with a link to that answer. There are also more links below this image.