Determine the number of solutions to this system of linear equations:

To determine the number of solutions to the given system of linear equations, we can analyze the equations graphically or algebraically.

The two equations are:

1. \( y = x + 5 \)
2. \( y = -x + 5 \)

We can rewrite these equations in standard form for clarity:

1. \( x - y + 5 = 0 \)
2. \( x + y - 5 = 0 \)

Next, we can solve the system algebraically by setting the two expressions for \(y\) equal to each other:

\[ x + 5 = -x + 5 \]

Now, we can solve for \(x\): \[ x + x = 5 - 5 \] \[ 2x = 0 \] \[ x = 0 \]

Now that we have \(x\), we can substitute it back into either of the original equations to find \(y\). Using the first equation:

\[ y = 0 + 5 = 5 \]

Thus, we find the solution is: \[ (x, y) = (0, 5) \]

To determine the number of solutions, we can also analyze the nature of the lines represented by the two equations. The first equation \(y = x + 5\) is a line with a slope of 1, and the second equation \(y = -x + 5\) is a line with a slope of -1.

Since the slopes of the two lines are different, they intersect at exactly one point. Therefore, the system of equations has one unique solution.

Conclusion: The number of solutions to the system of equations is \( \boxed{1} \).

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.

To analyze the statement about the system of equations:

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

We will first express both equations in a form that can be easily graphed and examine their lines.

Equation 1: The first equation, \( y = -x + 3 \), has:

• y-intercept of 3 (point (0, 3))
• slope of -1 (meaning it decreases one unit in \(y\) for each unit it increases in \(x\))

Equation 2: The second equation can be rearranged to solve for \(y\): \[ y = 8 - x \] This equation has:

• y-intercept of 8 (point (0, 8))
• slope of -1 (similar to the first equation)

Now, we can observe both equations:

• \( y = -x + 3 \): The line intersects the y-axis at (0, 3) and has a downward slope.
• \( y = 8 - x \): The line intersects the y-axis at (0, 8) and also has a downward slope.

Both lines have the same slope of -1, which means they are parallel. Since parallel lines never intersect, they do not share any points in common.

Therefore, these two equations do not have a solution: they are inconsistent.

Conclusion: The statement "Solving the system \( y = -x + 3 \) and \( x + y = 8 \) by graphing shows that there is no solution to the system" is true. The two lines representing these equations are parallel, confirming that there is no point at which they intersect, indicating no solution exists.

Does the graph show the system of equations x + y = 2 and -x + y = 7? Should the lines for the system be parallel?

To determine the relationship between the two equations \(x + y = 2\) and \(-x + y = 7\), we can analyze their slopes and intercepts.

Step 1: Put equations in slope-intercept form

Equation 1: Starting with \(x + y = 2\), we can rearrange it to slope-intercept form (\(y = mx + b\)): \[ y = -x + 2 \] This equation has:

• Slope (\(m\)): \(-1\)
• Y-intercept (\(b\)): \(2\) (point: \((0, 2)\))

Equation 2: Next, rearranging \(-x + y = 7\) into slope-intercept form: \[ y = x + 7 \] This equation has:

• Slope (\(m\)): \(1\)
• Y-intercept (\(b\)): \(7\) (point: \((0, 7)\))

Step 2: Compare slopes

We see that the slopes are:

• For \(y = -x + 2\) (slope = -1)
• For \(y = x + 7\) (slope = 1)

Since the slopes are different \((-1 \neq 1)\), this indicates that the lines are not parallel and will intersect at some point.

Conclusion

The lines for the system \(x + y = 2\) and \(-x + y = 7\) should not be parallel; instead, they intersect at one point. Therefore, the claim that these lines are parallel is false. The graph should show the intersection of these two lines.

If you were to graph these two equations, you would see that they cross each other at a specific point, which represents the unique solution to the system.

The graph of x + y = 2 is incorrect. The lines should intersect?

Let’s clarify and examine the equations \(x + y = 2\) and \(-x + y = 7\) carefully to determine their graphical representation, including their intersections.

Step 1: Convert equations to slope-intercept form.

1. For the equation \(x + y = 2\): \[ y = -x + 2 \]

   • Slope: \(-1\)
   • Y-intercept: \(2\) (point: \((0, 2)\))

2. For the equation \(-x + y = 7\): \[ y = x + 7 \]

   • Slope: \(1\)
   • Y-intercept: \(7\) (point: \((0, 7)\))

Step 2: Analyze the slopes

• The slope of the first line \(y = -x + 2\) is \(-1\) (a downward slope).
• The slope of the second line \(y = x + 7\) is \(1\) (an upward slope).

Since the slopes are different \((-1\) and \(1\)), the lines are not parallel and will intersect at some point.

Step 3: Find the point of intersection

To find the intersection point, we can solve the system of equations:

1. Start with the two equations: \[ x + y = 2 \quad (1) \] \[ -x + y = 7 \quad (2) \]

2. We can express \(y\) from equation (1): \[ y = 2 - x \]

3. Substitute this expression for \(y\) into equation (2): \[ -x + (2 - x) = 7 \] Simplifying this: \[ -x + 2 - x = 7 \] \[ -2x + 2 = 7 \] \[ -2x = 7 - 2 \] \[ -2x = 5 \] \[ x = -\frac{5}{2} \]

4. Now substitute \(x = -\frac{5}{2}\) back into equation (1) to find \(y\): \[ y = 2 - (-\frac{5}{2}) = 2 + \frac{5}{2} = \frac{4}{2} + \frac{5}{2} = \frac{9}{2} \]

Conclusion

The intersection point of the lines is \(\left(-\frac{5}{2}, \frac{9}{2}\right)\).

• Thus, the two lines do indeed intersect, which confirms that the graph of \(x + y = 2\) was not represented incorrectly in the assertion.
• If you have seen a graph where it seems the lines do not intersect, it would be the result of an error in the graph or its interpretation.

Final Confirmation: The lines of the system \(x + y = 2\) and \(-x + y = 7\) do intersect at the point \(\left(-\frac{5}{2}, \frac{9}{2}\right)\). Therefore, the statement that the lines should intersect is indeed true.

Choose
1. The graph of x + y = 2 is incorrect. The lines should intersect.
2. The graph of -x + y = 2 is incorrect. The lines should intersect.

Based on your question, we need to determine which statement correctly addresses the equations we have been discussing.

1. The graph of \(x + y = 2\) is incorrect. The lines should intersect.
2. The graph of \(-x + y = 2\) is incorrect. The lines should intersect.

From our previous conversation, we analyzed the system of equations:

1. \(x + y = 2\)
2. \(-x + y = 7\)

Summary of Findings:

• The first equation, \(x + y = 2\), can be rewritten as \(y = -x + 2\).
• The second equation, \(-x + y = 7\), can be rewritten as \(y = x + 7\).
• These two lines have different slopes (\(-1\) and \(1\)), which means they are not parallel and will intersect at a point.

Choosing the Correct Statement:

Considering the analysis above:

• The graph of \(x + y = 2\) is indeed correct, and the lines do intersect. Therefore, the first statement is false.

• The second statement mentions \(-x + y = 2\) which was never part of our original discussion; however, it likely refers to another equation. In context, \(-x + y = 7\) is the equation we analyzed, and it also has a different slope, meaning it is correct that this line will intersect with \(x + y = 2\). Therefore, if we treat \(-x + y = 2\) as an incorrect assumption regarding parallelism, we can suggest that the second statement is also false.

Since both statements appear to have factual inaccuracies based on the original equations discussed, if you are asking which one to choose considering intersection, the first statement is correct in identifying that the earlier graph represents lines that do intersect. However, this would be under the assumption that the first equation (\(x + y = 2\)) was misrepresented rather than indicating that \(x + y = 2\) itself is incorrect. So I would select:

1. The graph of \(x + y = 2\) is incorrect. The lines should intersect. (as a descriptor of confirming that a misunderstanding may have occurred regarding intersection).

However, be aware that this reflection is based on the details discussed, and if there was a specific graphical representation in question which clearly depicted \(x + y = 2\) inaccurately, selecting option 1 would align with requiring clarification.