How many solutions does the system of equations have? y=6x+2 and 3y-18x=12
A. one
B. two
C. infinitely many
D. none
the lines are parallel, they never cross. No solution
so it is none
To determine the number of solutions for the system of equations, we can compare the slopes of the two lines.
The given equations are:
1) y = 6x + 2
2) 3y - 18x = 12
The first equation is already in slope-intercept form (y = mx + b), where m represents the slope. From this equation, we can see that the slope is 6.
To convert the second equation into slope-intercept form, we isolate y:
3y - 18x = 12
3y = 18x + 12
y = (18x + 12)/3
y = 6x + 4
From this equation, we can see that the slope is also 6.
Since the slopes of the two lines are the same, the lines are parallel. When two lines are parallel, they never intersect, which means there are no solutions.
Therefore, the answer is:
D. none
To determine the number of solutions for the system of equations, we can solve them simultaneously. Here are the steps to solve the system of equations:
Step 1: Start by rewriting the second equation in terms of a single variable. Let's isolate y in the second equation:
3y - 18x = 12
3y = 18x + 12
y = 6x + 4
Now, we have two equations:
Equation 1: y = 6x + 2
Equation 2: y = 6x + 4
Step 2: Since both equations have the same slope (6), but different y-intercepts (2 and 4), they represent parallel lines.
Step 3: Since parallel lines never intersect, there are no common solutions to the system of equations.
Therefore, the correct answer is D. None.