Let's go through each question step by step to determine the correct answers.
First Question
Determine the number of solutions to this system of linear equations:
- \( y = x + 5 \)
- \( y = -x + 5 \)
To find the solutions:
- Set the two equations equal to find the intersection:
\( x + 5 = -x + 5 \)
Solving gives:
\( 2x = 0 \)
\( x = 0 \)
Substituting \( x = 0 \) back into either equation gives \( y = 5 \).
Thus, there is 1 unique solution.
Response:
Second Question
Is this statement true or false?
- "Solving the system \( y = -x + 3 \) and \( x + y = 8 \) by graphing shows that there is no solution to the system."
First, analyze the equations:
- \( y = -x + 3 \)
- \( x + y = 8 \) → can be rewritten as \( y = -x + 8 \).
Since both equations have the same slope of -1 (they are both in the form \( y = -x + b \)), they are parallel and will not intersect.
Response:
- The statement is true, because the lines are parallel.
Third Question
Does the graph show the system of equations \( x + y = 2 \) and \( -x + y = 7 \)? Should the lines for the system be parallel?
- Convert the equations:
- \( x + y = 2 \) → \( y = -x + 2 \)
- \( -x + y = 7 \) → \( y = x + 7 \)
These two equations have different slopes, meaning they will intersect at one point. Therefore, they shouldn't be parallel.
Response:
- The graph of \( -x + y = 7 \) is incorrect. The lines should intersect.
Fourth Question
Which of these graphs shows that the linear system \( y = -x + 6 \) and \( 3x + 3y = 18 \) has an infinite number of solutions?
The second equation can be simplified:
\( 3x + 3y = 18 \) → divide through by 3:
\( x + y = 6 \) → \( y = -x + 6 \)
This means both equations are the same line.
To have an infinite number of solutions, the graphs need to overlap completely. Look for two graphs that are the same line (one should be a solid line, while the other is a dotted line).
Response:
- A coordinate plane with 4 quadrants shows x and y axes ranging from negative 10 to 10 in unit increments. Two parallel lines are plotted on the plane. A solid downward slanting line with arrows at both ends passes through the points left parenthesis 0 comma 6 right parenthesis and left parenthesis 6 comma 0 right parenthesis. A dotted downward slanting line with arrows at both ends passes through the points left parenthesis 0 comma 6 right parenthesis and left parenthesis 6 comma 0 right parenthesis. (second from the top in text description)
You can check the graphs based on the calculated solutions and descriptions. If you have images to refer to, please cross-verify the specific graphical depictions mentioned here. If you need any more assistance or clarifications, feel free to ask!