Match the system of equations to the number of solutions it has.
y=5x+7
3y−15x=18
x−2y=6
3x−6y=18
y = 3x + 6
y=−13x−4
y=23x−1
y=23x−2
Infinite Solutions No Solutions One Solution
AAAaannndd the bot gets it wrong yet again!
y=5x+7
3y−15x=18
is equivalent to
y=5x+7
y=5x+6
so there are no solutions
x−2y=6
3x−6y=18
is equivalent to
x−2y=6
x−2y=6
so infinitely many solutions
The last two are obviously 1 solution and no solutions
My apologies for the mistake. Thank you for correcting me.
To determine the number of solutions for each system of equations, we can analyze the coefficients of the variables and the constants.
1. y=5x+7 and 3y−15x=18:
Since the second equation can be simplified to 3y-15x=18, we can divide all terms by 3 to get y-5x=6. This equation is equivalent to the first equation. Thus, the two equations are the same line and intersect at infinite points. This system has infinite solutions.
2. x−2y=6 and 3x−6y=18:
We can simplify the second equation by dividing all terms by 3 to get x-2y=6. Notice that this equation is equivalent to the first equation. Thus, the two equations represent the same line and intersect at infinite points. This system also has infinite solutions.
3. y = 3x + 6 and y=−13x−4:
Comparing the coefficients of x in both equations, we see that they are different (3 and -13). Since the lines represented by these equations have different slopes, they will intersect at one point. Thus, this system has one solution.
4. y=23x−1 and y=23x−2:
Both equations have the same coefficient for both x and y terms and the same constant term, which means they represent the same line. These equations are equivalent and intersect at infinite points. Therefore, this system has infinite solutions.