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Sagot :
Let’s examine each pair of linear equations to determine if they are consistent or inconsistent. If consistent, we will proceed to solve them graphically.
### (i) [tex]\( x + y = 5 \)[/tex], [tex]\( 2x + 2y = 10 \)[/tex]
First, we can recognize that the second equation is simply a multiple of the first equation:
[tex]\[ 2(x + y) = 2 \cdot 5 \][/tex]
[tex]\[ 2x + 2y = 10 \][/tex]
These two equations are essentially the same line. Thus, they are consistent and represent the same line. Hence, there are infinitely many solutions, as every point on the line [tex]\( x + y = 5 \)[/tex] is a solution.
### (ii) [tex]\( x - y = 8 \)[/tex], [tex]\( 3x - 3y = 16 \)[/tex]
We simplify the second equation by dividing by 3:
[tex]\[ \frac{3x - 3y}{3} = \frac{16}{3} \][/tex]
[tex]\[ x - y = \frac{16}{3} \][/tex]
Comparing with the first equation [tex]\( x - y = 8 \)[/tex], we see that:
[tex]\[ x - y = 8 \][/tex]
[tex]\[ x - y = \frac{16}{3} \][/tex]
Since [tex]\( 8 \neq \frac{16}{3} \)[/tex], these two lines are parallel and distinct, which means there is no point of intersection. Hence, the equations are inconsistent.
### (iii) [tex]\( 2x - y - 6 = 0 \)[/tex], [tex]\( 4x - 2y - 4 = 0 \)[/tex]
First, we simplify the second equation by dividing by 2:
[tex]\[ \frac{4x - 2y - 4}{2} = 0 \][/tex]
[tex]\[ 2x - y - 2 = 0 \][/tex]
Now we compare the two equations:
[tex]\[ 2x - y - 6 = 0 \][/tex]
[tex]\[ 2x - y - 2 = 0 \][/tex]
Rewriting these:
[tex]\[ 2x - y = 6 \][/tex]
[tex]\[ 2x - y = 2 \][/tex]
Since [tex]\( 6 \neq 2 \)[/tex], these two equations represent parallel lines, indicating no point of intersection. Hence, the equations are inconsistent.
### (iv) [tex]\( 2x - 2y - 2 = 0 \)[/tex], [tex]\( 4x - 4y - 5 = 0 \)[/tex]
First, we simplify the first equation:
[tex]\[ \frac{2x - 2y - 2}{2} = 0 \][/tex]
[tex]\[ x - y - 1 = 0 \][/tex]
[tex]\[ x - y = 1 \][/tex]
Next, we simplify the second equation:
[tex]\[ \frac{4x - 4y - 5}{4} = 0 \][/tex]
[tex]\[ x - y - \frac{5}{4} = 0 \][/tex]
[tex]\[ x - y = \frac{5}{4} \][/tex]
Since [tex]\( 1 \neq \frac{5}{4} \)[/tex], these represent parallel lines, indicating no point of intersection. Hence, the equations are inconsistent.
### Summary:
- (i) Consistent, with infinitely many solutions (same line).
- (ii) Inconsistent (parallel and distinct).
- (iii) Inconsistent (parallel and distinct).
- (iv) Inconsistent (parallel and distinct).
For the consistent system in (i):
We can graphically solve [tex]\( x + y = 5 \)[/tex].
1. Find two points that satisfy the equation, for instance:
- If [tex]\( x = 0 \)[/tex]: [tex]\( y = 5 \)[/tex], so (0, 5).
- If [tex]\( y = 0 \)[/tex]: [tex]\( x = 5 \)[/tex], so (5, 0).
2. Plot these points on a coordinate plane:
- Point [tex]\( (0, 5) \)[/tex]
- Point [tex]\( (5, 0) \)[/tex]
3. Draw a line through these points. Every point on this line is a solution to the system.
The graphical representation will show a line with an infinite number of solutions where [tex]\( x + y = 5 \)[/tex].
### (i) [tex]\( x + y = 5 \)[/tex], [tex]\( 2x + 2y = 10 \)[/tex]
First, we can recognize that the second equation is simply a multiple of the first equation:
[tex]\[ 2(x + y) = 2 \cdot 5 \][/tex]
[tex]\[ 2x + 2y = 10 \][/tex]
These two equations are essentially the same line. Thus, they are consistent and represent the same line. Hence, there are infinitely many solutions, as every point on the line [tex]\( x + y = 5 \)[/tex] is a solution.
### (ii) [tex]\( x - y = 8 \)[/tex], [tex]\( 3x - 3y = 16 \)[/tex]
We simplify the second equation by dividing by 3:
[tex]\[ \frac{3x - 3y}{3} = \frac{16}{3} \][/tex]
[tex]\[ x - y = \frac{16}{3} \][/tex]
Comparing with the first equation [tex]\( x - y = 8 \)[/tex], we see that:
[tex]\[ x - y = 8 \][/tex]
[tex]\[ x - y = \frac{16}{3} \][/tex]
Since [tex]\( 8 \neq \frac{16}{3} \)[/tex], these two lines are parallel and distinct, which means there is no point of intersection. Hence, the equations are inconsistent.
### (iii) [tex]\( 2x - y - 6 = 0 \)[/tex], [tex]\( 4x - 2y - 4 = 0 \)[/tex]
First, we simplify the second equation by dividing by 2:
[tex]\[ \frac{4x - 2y - 4}{2} = 0 \][/tex]
[tex]\[ 2x - y - 2 = 0 \][/tex]
Now we compare the two equations:
[tex]\[ 2x - y - 6 = 0 \][/tex]
[tex]\[ 2x - y - 2 = 0 \][/tex]
Rewriting these:
[tex]\[ 2x - y = 6 \][/tex]
[tex]\[ 2x - y = 2 \][/tex]
Since [tex]\( 6 \neq 2 \)[/tex], these two equations represent parallel lines, indicating no point of intersection. Hence, the equations are inconsistent.
### (iv) [tex]\( 2x - 2y - 2 = 0 \)[/tex], [tex]\( 4x - 4y - 5 = 0 \)[/tex]
First, we simplify the first equation:
[tex]\[ \frac{2x - 2y - 2}{2} = 0 \][/tex]
[tex]\[ x - y - 1 = 0 \][/tex]
[tex]\[ x - y = 1 \][/tex]
Next, we simplify the second equation:
[tex]\[ \frac{4x - 4y - 5}{4} = 0 \][/tex]
[tex]\[ x - y - \frac{5}{4} = 0 \][/tex]
[tex]\[ x - y = \frac{5}{4} \][/tex]
Since [tex]\( 1 \neq \frac{5}{4} \)[/tex], these represent parallel lines, indicating no point of intersection. Hence, the equations are inconsistent.
### Summary:
- (i) Consistent, with infinitely many solutions (same line).
- (ii) Inconsistent (parallel and distinct).
- (iii) Inconsistent (parallel and distinct).
- (iv) Inconsistent (parallel and distinct).
For the consistent system in (i):
We can graphically solve [tex]\( x + y = 5 \)[/tex].
1. Find two points that satisfy the equation, for instance:
- If [tex]\( x = 0 \)[/tex]: [tex]\( y = 5 \)[/tex], so (0, 5).
- If [tex]\( y = 0 \)[/tex]: [tex]\( x = 5 \)[/tex], so (5, 0).
2. Plot these points on a coordinate plane:
- Point [tex]\( (0, 5) \)[/tex]
- Point [tex]\( (5, 0) \)[/tex]
3. Draw a line through these points. Every point on this line is a solution to the system.
The graphical representation will show a line with an infinite number of solutions where [tex]\( x + y = 5 \)[/tex].
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