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Solving the system y = -x + 3 and x + y = 8 by graphing shows that there is no solution to the system. Is this statement true or false? Explain.

answers:

The statement is true, because the lines have an intersection point.

The statement is false, because the lines have an intersection point

The statement is false, because the lines are parallel

The statement is true, because the lines are parallel

3 answers

To determine whether the statement is true or false, we first need to analyze the two equations given:

  1. \( y = -x + 3 \)
  2. \( x + y = 8 \)

Let's rewrite the second equation in slope-intercept form (y = mx + b):

  1. From \( x + y = 8 \), we can solve for \( y \): \[ y = -x + 8 \]

Now we have the two equations:

  1. \( y = -x + 3 \)
  2. \( y = -x + 8 \)

We can see that both equations have the same slope (-1), which indicates that they are parallel lines. Since parallel lines do not intersect, they do not have any solution in terms of a point of intersection.

Now we can evaluate the statement:

"The statement is true, because the lines are parallel."

Thus, the correct answer is:

The statement is true, because the lines are parallel.

you sure

Yes, let's carefully revisit the problem to ensure clarity.

We have the two equations:

  1. \( y = -x + 3 \)
  2. \( x + y = 8 \)

First, we rewrite the second equation in slope-intercept form:

From \( x + y = 8 \), we isolate \( y \):

\[ y = -x + 8 \]

Now we compare the two equations in slope-intercept form:

  1. \( y = -x + 3 \) (slope = -1, y-intercept = 3)
  2. \( y = -x + 8 \) (slope = -1, y-intercept = 8)

Both lines have the same slope (\(-1\)), but different y-intercepts (3 and 8), which means they are parallel lines. Since parallel lines never intersect, there is no solution for this system of equations.

Thus, the initial assessment holds:

The statement is true, because the lines are parallel.

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