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The point (0,-2) is a solution to which system?

options:

A). x + y = 2 and -x + 2y = 16

B). -5x + y = -2 and -3x + 6y = -12

C). -4x + y = 6 and -5x - y = 21

D). -5x = y - 3 and 3x - 8y = 24

11 answers

To determine which system has the point (0, -2) as a solution, we will substitute \(x = 0\) and \(y = -2\) into each equation of the provided options. If both equations of a system are satisfied, then (0, -2) is a solution for that system.

Option A:

  1. \(x + y = 2\)
    Substitute: \(0 + (-2) = 2\)
    Result: \(-2 \neq 2\) (not satisfied)

  2. \(-x + 2y = 16\)
    Substitute: \(-0 + 2(-2) = 16\)
    Result: \(-4 \neq 16\) (not satisfied)

Conclusion for A: Not a solution.

Option B:

  1. \(-5x + y = -2\)
    Substitute: \(-5(0) + (-2) = -2\)
    Result: \(-2 = -2\) (satisfied)

  2. \(-3x + 6y = -12\)
    Substitute: \(-3(0) + 6(-2) = -12\)
    Result: \(-12 = -12\) (satisfied)

Conclusion for B: Is a solution.

Option C:

  1. \(-4x + y = 6\)
    Substitute: \(-4(0) + (-2) = 6\)
    Result: \(-2 \neq 6\) (not satisfied)

  2. \(-5x - y = 21\)
    Substitute: \(-5(0) - (-2) = 21\)
    Result: \(2 \neq 21\) (not satisfied)

Conclusion for C: Not a solution.

Option D:

  1. \(-5x = y - 3\)
    Substitute: \(-5(0) = -2 - 3\)
    Result: \(0 \neq -5\) (not satisfied)

  2. \(3x - 8y = 24\)
    Substitute: \(3(0) - 8(-2) = 24\)
    Result: \(16 \neq 24\) (not satisfied)

Conclusion for D: Not a solution.

After evaluating all options, we find that the point (0, -2) is a solution to Option B: \(-5x + y = -2\) and \(-3x + 6y = -12\).

Match the system of equations to the number of solutions it has.

y = 5x + 7

3y - 15x = 18

To find the number of solutions for the system of equations, we can analyze the given equations:

  1. Equation 1:
    \(y = 5x + 7\)
    This is in slope-intercept form \(y = mx + b\), where the slope \(m = 5\) and the y-intercept \(b = 7\).

  2. Equation 2:
    \(3y - 15x = 18\)
    We can rearrange this into slope-intercept form. First, solve for \(y\):

    \[ 3y = 15x + 18 \] Now, divide by 3: \[ y = 5x + 6 \]

Now we have both equations in slope-intercept form:

  1. \(y = 5x + 7\) (slope = 5, y-intercept = 7)
  2. \(y = 5x + 6\) (slope = 5, y-intercept = 6)

Analysis of Solutions:

  • Both lines have the same slope (5), which means they are parallel.
  • Since they have different y-intercepts (7 and 6), they are distinct lines that will never intersect.

Conclusion:

Since the two equations represent parallel lines, the system of equations has no solutions.

y = 3x + 6

y = -13x - 4

To determine the number of solutions for the system of equations given by:

  1. \(y = 3x + 6\)
  2. \(y = -13x - 4\)

we can analyze the slopes and intercepts of both equations.

Step 1: Identify the slopes and y-intercepts

  1. For the first equation \(y = 3x + 6\):

    • Slope (\(m_1\)) = 3
    • Y-intercept (\(b_1\)) = 6

  2. For the second equation \(y = -13x - 4\):

    • Slope (\(m_2\)) = -13
    • Y-intercept (\(b_2\)) = -4

Step 2: Compare the slopes

  • The slope of the first line (\(m_1 = 3\)) is different from the slope of the second line (\(m_2 = -13\)).
  • Since the slopes are different, the two lines are not parallel and must intersect at exactly one point.

Conclusion

Since the lines intersect at one point, the system of equations has one solution.

x - 2y = 6

3x - 6y = 18

To determine the number of solutions for the system of equations given by:

  1. \(x - 2y = 6\)
  2. \(3x - 6y = 18\)

we can analyze these equations.

Step 1: Simplify the second equation

The second equation can be simplified. Notice that:

\[ 3x - 6y = 18 \]

We can divide all terms by 3:

\[ x - 2y = 6 \]

Step 2: Compare the equations

Now we have:

  1. \(x - 2y = 6\) (Equation 1)
  2. \(x - 2y = 6\) (Equation 2)

Both equations are identical.

Conclusion

Since both equations represent the same line, there are infinitely many solutions. In this case, every point on the line \(x - 2y = 6\) is a solution to the system. Thus, the system has infinitely many solutions.

y = 23x - 1

y = 23x - 2

To determine the number of solutions for the system of equations given by:

  1. \(y = 23x - 1\)
  2. \(y = 23x - 2\)

Step 1: Identify the slopes and y-intercepts

  • For the first equation \(y = 23x - 1\):

    • Slope (\(m_1\)) = 23
    • Y-intercept (\(b_1\)) = -1

  • For the second equation \(y = 23x - 2\):

    • Slope (\(m_2\)) = 23
    • Y-intercept (\(b_2\)) = -2

Step 2: Compare the slopes

  • Both lines have the same slope (\(m_1 = m_2 = 23\)), indicating that they are parallel.
  • However, they have different y-intercepts (-1 and -2), which means they will never intersect.

Conclusion

Since the two lines are parallel and do not intersect, the system of equations has no solutions.

Judy has $100 and Elenore has $75. Judy is saving $5 per day and Elenore is saving $7 per day.

options:

What equation represents how much money Judy will have in total after saving for d days?

To represent how much money Judy will have in total after saving for \(d\) days, we need to consider her initial amount of money and how much she saves each day.

  1. Initial amount Judy has: $100
  2. Amount Judy saves per day: $5

After \(d\) days, the total amount of money Judy will have can be calculated as follows:

\[ \text{Total amount} = \text{Initial amount} + (\text{Savings per day} \times \text{Number of days}) \]

Substituting the values:

\[ \text{Total amount} = 100 + 5d \]

Thus, the equation that represents how much money Judy will have in total after saving for \(d\) days is:

\[ y = 100 + 5d \]

where \(y\) is the total amount of money Judy has after \(d\) days.