How many solutions does the system of equations have?
x-3y=15 and 3x-9y=5
one
two
infinitely many
none
How many solutions does the system of equations have?
y-5x= -6 and 3y-15x= -12
one
two
infinitely many
none
How many solutions are there to this problem;9y + 5 = -5 + 4y?
For the first system of equations, let's have some fun with math and comedy! It's like juggling a few equations. So, let's see what's going on here. We have:
x-3y=15 and 3x-9y=5
Now, what happens if we multiply the first equation by 3? We get:
3x-9y=45
Oh no, we have conflicting results! The second equation says 3x-9y=5, but the modified first equation says 3x-9y=45. These two results don't match, which means we have no solution! Just like a joke without a punchline, there's no answer here.
Now, moving on to the second system of equations:
y-5x= -6 and 3y-15x= -12
Let's simplify the second equation by multiplying it by -1:
-3y+15x=12
Ah, much better! Now we see a twist in the equation, where the first equation is y-5x= -6 and the modified second equation is -3y+15x=12. These are actually the same equation, just written differently! In other words, they are like a pair of clowns wearing the same outfit. And just like the two clowns, these equations have infinitely many solutions. So, the answer is infinitely many, just like the number of jokes we can come up with about clowns!
To determine the number of solutions in a system of equations, we can use the concept of slopes.
For the first system:
Equation 1: x - 3y = 15
Equation 2: 3x - 9y = 5
First, we can simplify Equation 1 and 2 by dividing both sides of each equation by a common factor:
Equation 1: (1/3)x - y = 5
Equation 2: (1/3)x - y = 5/3
We can see that the two equations have the same slope (-1) and the same y-intercept (5/3). This means that the two lines are parallel and will never intersect. Therefore, the system has no solution.
For the second system:
Equation 1: y - 5x = -6
Equation 2: 3y - 15x = -12
We can simplify Equation 1 and 2 by dividing both sides of each equation by a common factor:
Equation 1: (1/5)y - x = -6/5
Equation 2: (1/5)y - x = -4/5
We can see that the two equations have the same slope (-1) and the same y-intercept (-4/5). This means that the two lines are identical and will coincide with each other. Therefore, the system has infinitely many solutions.
In summary:
- The first system has no solution.
- The second system has infinitely many solutions.
To determine the number of solutions for each system of equations, we can use the concept of equations in slope-intercept form (y = mx + b). By manipulating the given equations into this form, we can observe their slopes and y-intercepts to see if they have a unique solution or multiple solutions.
1) System of Equations: x - 3y = 15 and 3x - 9y = 5
Step 1: Convert the equations to slope-intercept form.
For equation 1: x - 3y = 15
Rearrange the equation to isolate y:
-3y = -x + 15
y = (1/3)x - 5
For equation 2: 3x - 9y = 5
Rearrange the equation to isolate y:
-9y = -3x + 5
y = (1/3)x - 5/9
In both equations, the coefficients of x and y are the same, and the y-intercept is different. This indicates that the two equations represent parallel lines.
Step 2: Determine the number of solutions.
When two lines are parallel, they do not intersect, which means there is no point of intersection and no solution. Therefore, the answer is "none."
2) System of Equations: y - 5x = -6 and 3y - 15x = -12
Step 1: Convert the equations to slope-intercept form.
For equation 1: y - 5x = -6
Rearrange the equation to isolate y:
y = 5x - 6
For equation 2: 3y - 15x = -12
Rearrange the equation to isolate y:
y = 5x - 4
In both equations, the coefficients of x and y are the same, and the y-intercepts are the same. This indicates that the two equations represent the same line.
Step 2: Determine the number of solutions.
Since the two equations are the same, they intersect at every point along the line they represent. Therefore, there are infinitely many solutions.
In conclusion:
1) The system of equations x - 3y = 15 and 3x - 9y = 5 has no solution.
2) The system of equations y - 5x = -6 and 3y - 15x = -12 has infinitely many solutions.
multiply the 1st by 3 and you have
3x-9y=45
3x-9y=5
There's no way they can both be true. The line sare parallel, so they do not intersect.
same for #2. Try it and see what happens.