Match the system of equations to the number of solutions it has.
One Solution, No Solution, Infinite Solutions
(4 points)
y = 5x + 7
3y - 15x = 18
y = 3x + 6
y = -13x - 4
x - 2y = 6
3x - 6y = 18
y = 23x - 1
y = 23x - 2
One Solution:
y = 5x + 7
3y - 15x = 18
Infinite Solutions:
y = 3x + 6
y = -13x - 4
No Solution:
x - 2y = 6
3x - 6y = 18
One Solution:
y = 23x - 1
y = 23x - 2
are you sure???
Apologies for the confusion. Let me revise my previous response:
One Solution:
- y = 5x + 7
- 3y - 15x = 18
No Solution:
- y = 3x + 6
- y = -13x - 4
Infinite Solutions:
- x - 2y = 6
- 3x - 6y = 18
One Solution:
- y = 23x - 1
- y = 23x - 2
Thank you for pointing out the mistake, and I apologize for any confusion caused.
To determine the number of solutions for each system, we need to first rewrite them in standard form (Ax + By = C), where A, B, and C are constants.
1. y = 5x + 7
3y - 15x = 18
To convert the first equation to standard form, we move all terms to one side:
-5x + y = 7
The second equation is already in standard form.
This system has one solution because it consists of two linear equations with different slopes, meaning the lines representing them intersect at a single point.
2. y = 3x + 6
y = -13x - 4
Both equations are already in slope-intercept form (y = mx + b).
This system also has one solution since the lines represented by the equations have different slopes and intersect at one point.
3. x - 2y = 6
3x - 6y = 18
We need to rearrange the equations to standard form:
x - 2y = 6 -> -2y = -x + 6 -> 2y = x - 6
3x - 6y = 18 -> -6y = -3x + 18 -> 6y = 3x - 18
Simplifying these equations, we get:
2y = x - 6 -> x - 2y = 6
6y = 3x - 18 -> x - 2y = 6
This system consists of two equations representing the same line. Therefore, it has infinite solutions since any point on the line will satisfy both equations.
4. y = 23x - 1
y = 23x - 2
Both equations have the same slope-intercept form and represent the same line.
Similar to system 3, this system also has infinite solutions since any point on the line will satisfy both equations.
In summary:
1. One Solution
2. One Solution
3. Infinite Solutions
4. Infinite Solutions