- y = 5x + 7
_____________
3y - 15x = 18
- y = 3x + 6
________________
y = -13x - 4
- x - 2y = 6
_____________________
3x - 6y = 18
- y = 23x - 1
_____________________
y = 23x - 2
- Options:
- One solution
- No solution
- Infinite solutions
- One solution
To determine whether each system of equations has one solution, no solution, or infinite solutions, we can use the method of solving systems of equations.
1. y = 5x + 7
3y - 15x = 18
To solve this system, we can substitute the value of y from the first equation into the second equation:
3(5x + 7) - 15x = 18
15x + 21 - 15x = 18
21 = 18
Since 21 is not equal to 18, the system has no solution.
Answer: No solution
2. y = 3x + 6
y = -13x - 4
Since both equations have the same value of y on one side, we can set them equal to each other:
3x + 6 = -13x - 4
3x + 13x = -4 - 6
16x = -10
x = -10/16
x = -5/8
Substituting the value of x back into either equation, we can find the value of y:
y = 3(-5/8) + 6
y = -15/8 + 48/8
y = 33/8
So, the system has one solution.
Answer: One solution
3. x - 2y = 6
3x - 6y = 18
To solve this system, we can multiply the first equation by 3 to eliminate x:
3(x - 2y) = 3(6)
3x - 6y = 18
This is exactly the same as the second equation, so the system has infinitely many solutions.
Answer: Infinite solutions
4. y = 23x - 1
y = 23x - 2
Since both equations have the same value of y on one side, we can set them equal to each other:
23x - 1 = 23x - 2
1 = 2
Since 1 is not equal to 2, the system has no solution.
Answer: No solution
In summary:
1. No solution
2. One solution
3. Infinite solutions
4. No solution
To determine the number of solutions for each system of equations, we need to compare the slopes and y-intercepts of the lines represented by the equations. Let's analyze each system of equations:
1. y = 5x + 7 (Equation A)
3y - 15x = 18 (Equation B)
In Equation A, the slope is 5, and the y-intercept is 7.
In Equation B, dividing both sides by 3, we get: y - 5x = 6.
So, the slope is 5, and the y-intercept is 6/5.
Since the slopes are equal (5 = 5), and the y-intercepts are different (7 ≠ 6/5), these lines are parallel. Therefore, there is no solution to this system of equations.
Answer: No solution.
2. y = 3x + 6 (Equation A)
y = -13x - 4 (Equation B)
Comparing the equations, we can see that both lines have different slopes and different y-intercepts.
Since the slopes and y-intercepts are different, these lines will intersect at a single point, thus having one solution.
Answer: One solution.
3. x - 2y = 6 (Equation A)
3x - 6y = 18 (Equation B)
To make the analysis easier, let's multiply Equation A by 3, resulting in:
3x - 6y = 18 (Equation A)
Now, we have:
3x - 6y = 18 (Equation A)
3x - 6y = 18 (Equation B)
Both equations are identical. This implies that any values of x and y that satisfy one equation will also satisfy the other. Therefore, these equations are equivalent and have infinite solutions.
Answer: Infinite solutions.
4. y = 23x - 1 (Equation A)
y = 23x - 2 (Equation B)
Comparing the equations, we can see that both lines have the same slope (23) but different y-intercepts (-1 ≠ -2). Since the lines have the same slope but different y-intercepts, they will never intersect. Therefore, this system of equations has no solution.
Answer: No solution.
In summary, the systems of equations have the following solutions:
1. No solution
2. One solution
3. Infinite solutions
4. No solution