The graphs represent this system: y=3x−2 −3x+y=7 How many solutions does this system have: 0, 1, or ∞ ? (1 point)
The system of equations is:
y = 3x - 2
-3x + y = 7
We can solve this system by substitution or elimination method.
Using the elimination method, we can add 3x to both sides of the second equation:
-2x + y = 7
Now, we can see that the first equation and the modified second equation are equivalent.
This means that the system of equations represents the same line.
Since the two equations represent the same line, the system has infinitely many solutions.
Therefore, the answer is ∞.
Wrong
Apologies for the mistake in my previous response.
To determine the number of solutions, we need to analyze the two equations.
The first equation, y = 3x - 2, represents a straight line with a slope of 3 and a y-intercept of -2.
The second equation, -3x + y = 7, can be rewritten as y = 3x + 7. This equation also represents a straight line with a slope of 3 and a y-intercept of 7.
Since the slopes of both lines are equal, but the y-intercepts are different, these lines are parallel and will never intersect. Therefore, the system of equations has no solution.
The correct answer is 0.
To determine the number of solutions for the given system of equations, we can analyze the slopes of the lines represented by the equations.
The first equation, y = 3x - 2, has a slope of 3.
To find the slope of the second equation, we need to rewrite it in slope-intercept form (y = mx + b).
Rearranging the equation -3x + y = 7, we get y = 3x + 7.
Thus, the slope of the second equation is also 3.
Since the slopes of both equations are equal, the lines are parallel.
When two lines are parallel, they will never intersect, meaning there are no common solutions for the system of equations.
Therefore, the system y = 3x - 2 and -3x + y = 7 has 0 solutions.
To determine the number of solutions in a system of equations, we need to solve the equations simultaneously. Let's solve the given system of equations:
y = 3x - 2 ...(Equation 1)
-3x + y = 7 ...(Equation 2)
We can use either the substitution method or the elimination method to solve this system. Here, we will use the elimination method.
Step 1: Multiply Equation 1 by -1:
-1(y) = -1(3x - 2)
-y = -3x + 2 ...(Equation 3)
Step 2: Add Equations 2 and 3 together:
-3x + y + (-3x + 2) = 7 ± 0
-3x + y - 3x + 2 = 7
-6x + y + 2 = 7
y - 6x = 7 - 2
y - 6x = 5 ...(Equation 4)
Now we have a new equation, Equation 4, which represents the same line as the original system.
Step 3: Compare Equations 1 and 4:
We can see that Equation 4 is simply Equation 1 rewritten in a different form, as both represent the line y = 3x - 2.
Since both equations represent the same line, there are infinitely many solutions to this system. The system is consistent and dependent.
Therefore, the correct answer is ∞ (infinity).