The point (5,-3) is a solution to which system of equations? (1 point)

Question

A. y=x-8 2x+y=7
B. y=2/3x+6 3y+6x-18=0
C. y=x+2 y=x+5
D. y=-1/2x+6 y=3x-1

Answer 1

To determine which system of equations has the point (5, -3) as a solution, we need to substitute $x=5$ and $y=-3$ into each equation in the systems provided and see which system holds true for both equations.

Option A:

1. Equation 1: $y=x-8$
   Substitute $x=5$:
   $y=5-8=-3$ (True)

2. Equation 2: $2x+y=7$
   Substitute $x=5$ and $y=-3$:
   $2(5)+(-3)=10-3=7$ (True)

Both equations in Option A are satisfied.

Option B:

1. Equation 1: $y=\frac{2}{3}x+6$
   Substitute $x=5$:
   $y=\frac{2}{3}(5)+6=\frac{10}{3}+6=\frac{10}{3}+\frac{18}{3}=\frac{28}{3}$ (Not true, since -3 is not equal to $\frac{28}{3}$)

Since the first equation is not satisfied, we can discard this option.

Option C:

1. Equation 1: $y=x+2$
   Substitute $x=5$:
   $y=5+2=7$ (Not true since -3 is not equal to 7)

Again, since the first equation is not satisfied, we can discard this option.

Option D:

1. Equation 1: $y=-\frac{1}{2}x+6$
   Substitute $x=5$:
   $y=-\frac{1}{2}(5)+6=-\frac{5}{2}+6=-\frac{5}{2}+\frac{12}{2}=\frac{7}{2}$ (Not true since -3 is not equal to $\frac{7}{2}$)

Since the first equation is not satisfied, we can discard this option.

Conclusion:

After checking all options, the only system that satisfies the point (5, -3) is:

Option A: $y=x-8$ and $2x+y=7$

Answer 2

How many solutions does the system of equations have?
y=-2x+12
4y+x-20=0 (1 point)
A. Infinite solutions
B. One solution
C. Two solutions
D. No solution

Answer 3

To determine how many solutions the system of equations has, we can analyze the two equations given:

1. Equation 1: $y=-2x+12$

2. Equation 2: $4y+x-20=0$

First, let's rewrite Equation 2 in slope-intercept form (i.e., $y=mx+b$):

Starting with: $$4y+x-20=0$$

We can isolate $y$: $$4y=-x+20$$

Now, divide by 4: $$y=-\frac{1}{4}x+5$$

Now, we have both equations in the form of $y=mx+b$:

1. $y=-2x+12$ (slope = -2)
2. $y=-\frac{1}{4}x+5$ (slope = -1/4)

Analysis

• The slopes of the two lines are different (-2 and -1/4).
• Since the slopes are not equal, the lines will intersect at one point.

Conclusion

The system of equations has one solution.

The correct answer is: B. One solution.