Discover the best answers at Westonci.ca, where experts share their insights and knowledge with you. Join our Q&A platform to connect with experts dedicated to providing precise answers to your questions in different areas. Join our Q&A platform to connect with experts dedicated to providing accurate answers to your questions in various fields.
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].
We appreciate your time on our site. Don't hesitate to return whenever you have more questions or need further clarification. Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. We're glad you chose Westonci.ca. Revisit us for updated answers from our knowledgeable team.
