**Solving special systems by graphing :**

When we graph the system of two linear equations, we will get two straight lines.

If the lines intersect each other, there is a solution for the given system.

If the lines never intersect, there is no solution for the given system.

**Example 1 : **

Solve the given system of equations by graphing.

x + y = 7

4x + 4y = 12

**Solution :**

**Step 1 :**

Write the given equations in slope-intercept form.

**y = mx + b **

x + y = 7

y = - x + 7

(slope is -1 and y-intercept is 7)

4x + 4y = 12

Divide both sides by 4.

x + y = 3

y = - x + 3

(slope is -1 and y-intercept is 3)

Based on slope and y-intercept, we can graph the given equations.

**Step 2 : **

In the above graph, the lines do not intersect. They appear to be parallel and and no points in common.

Does this linear system have a solution ? Use the graph to explain.

Since the lines are parallel and they do not intersect, there is no point of intersection. So, the system has no solution.

**Reflection : **

1. Can you conclude that the system of linear equation has no solution without graphing ?

Yes

We have to find the slopes and y-intercepts of the given two lines. If the slopes are equal, but y-intercepts are different, the lines will be parallel. Then, the system will not have solution.

2. Can you conclude that the system of linear equation has infinitely many solutions without graphing ?

Yes

We have to find the slopes and y-intercepts of the given two lines. If the slopes are equal and y-intercepts are also equal, both the equations represent the same line. All ordered pairs on the line will make both equations true and all points on the line are points of intersection.

Then, the system will have infinitely many solutions.

**Example 2 :**

Determine whether the system given below has one solution, infinitely many solutions or no solution.

x + y = 3

2x + 2y = 6

**Solution :**

**Step 1 : **

Write the given equations in slope-intercept form.

**y = mx + b **

x + y = 3

y = - x + 3

(slope is -1 and y-intercept is 3)

2x + 2y = 6

Divide both sides by 2.

x + y = 3

y = - x + 3

(slope is -1 and y-intercept is 3)

**Step 2 :**

The slopes of both the equations are same and y-intercepts are also same.

So, both the equations represents the same line. All ordered pairs on the line will make both equations true and all points on the line are points of intersection.

Hence, the system has infinitely many solutions.

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