Is (−2, 6) a solution to the system of these linear equations: x+2y=10 and 3x+y=0 ? Why?(1 point) Responses No, because the graphs intersect at (−2, 6). No, because the graphs intersect at left parenthesis negative 2 comma 6 right parenthesis . Yes, because the graphs don’t intersect at (−2, 6). Yes, because the graphs don’t intersect at left parenthesis negative 2 comma 6 right parenthesis . Yes, because the graphs intersect at (−2, 6). Yes, because the graphs intersect at left parenthesis negative 2 comma 6 right parenthesis . No, because the graphs don’t intersect at (−2, 6).

Question

GPT-4o mini · Answer

To determine if the point $(-2, 6)$ is a solution to the system of equations $x + 2y = 10$ and $3x + y = 0$, we will substitute $x = -2$ and $y = 6$ into both equations. 1. Substitute into the first equation: $$ -2 + 2(6) = -2 + 12 = 10 $$ This is correct, as it satisfies the first equation. 2. Substitute into the second equation: $$ 3(-2) + 6 = -6 + 6 = 0 $$ This is also correct, as it satisfies the second equation. Since $(-2, 6)$ satisfies both equations, it is indeed a solution to the system of equations. Now, regarding the options provided: - The correct answer should be “Yes, because the graphs intersect at $(-2, 6)$.” This indicates that the point $(-2, 6)$ is a solution because it is the point where the two lines represented by the equations intersect.

...hi.... · Answer

Use the image to answer the question. Estimate the solution to the system of these linear equations based on the graph. y=5x+10 y=−2x+3 (1 point) Responses (−1,−5) left parenthesis negative 1 comma negative 5 right parenthesis (1, 5) left parenthesis 1 comma 5 right parenthesis (−1, 5) left parenthesis negative 1 comma 5 right parenthesis (5,−1) left parenthesis 5 comma negative 1 right parenthesis Skip to navigation

GPT-4o mini · Answer

To estimate the solution to the system of equations $y = 5x + 10$ and $y = -2x + 3$ based on a graph, you would typically look for the point of intersection of the two lines represented by these equations.

Graph $y = 5x + 10$:
- This line has a slope of 5 and a y-intercept of 10. It is steep and increases sharply as x increases.
Graph $y = -2x + 3$:
- This line has a slope of -2 and a y-intercept of 3. It decreases as x increases.

To find their intersection:

The x-coordinate of the intersection will be where both equations result in the same y-coordinate.

If you have the graph and can visually examine it, identify the point where the two lines intersect and then match this point with the given options.

Based on the possible options you've provided:

Depending on the graphical intersection, choose the option that corresponds to the coordinates of the intersection point.

Without the graph, I can’t definitively say which option is correct, but you should look for the coordinate that appears to be the intersection point in the graph.

If you can refer to the image to identify this point, that will lead you to the correct answer. If you provide me with a description or approximate coordinates based on the graph, I could help confirm the best choice!

...hi.... · Answer

What is the last step to solving a system of equations?(1 point) Responses Make a table of solutions of the linear equations. Make a table of solutions of the linear equations. Estimate the intersection point. Estimate the intersection point. Graph the lines. Graph the lines. Check the answer.

GPT-4o mini · Answer

The last step to solving a system of equations is typically to check the answer. This involves substituting the solution back into the original equations to ensure that it satisfies both equations.

So, the correct response is: Check the answer.

...hi.... · Answer

Determine the number of solutions to this system of linear equations: y=x+5 y=−x+5 (1 point) Responses 2 2 0 0 infinite infinite 1

GPT-4o mini · Answer

To determine the number of solutions to the system of linear equations:

$y = x + 5$
$y = -x + 5$

We can analyze the equations by attempting to find the point(s) where they intersect.

Set the equations equal to each other to find the intersection point: \[ x + 5 = -x + 5 \]
Solve for $x$: \[ x + x = 5 - 5 \] \[ 2x = 0 \quad \Rightarrow \quad x = 0 \]
Substitute $x = 0$ into either equation to find $y$: \[ y = 0 + 5 = 5 \]

So, the solution is $(0, 5)$.

Since we found a single point of intersection, this system has exactly one solution.

Thus, the correct response is: 1.

...hi.... · Answer

Use the image to answer the question. 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. (1 point) Responses The statement is true, because the lines have an intersection point. 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 have an intersection point. The statement is false, because the lines are parallel. The statement is false, because the lines are parallel. The statement is true, because the lines are parallel. The statement is true, because the lines are parallel. Skip to navigation

GPT-4o mini · Answer

To determine the truth of the statement regarding the system of equations $y = -x + 3$ and $x + y = 8$ regarding their solutions:

Rewrite the second equation $x + y = 8$ in slope-intercept form (y = mx + b): \[ y = -x + 8 \]

Now we have the two equations:

$y = -x + 3$
$y = -x + 8$

Analyze the slopes:
- Both equations have the same slope of -1, which indicates that they are parallel lines.
Determine the y-intercepts:
- The first line intersects the y-axis at 3.
- The second line intersects the y-axis at 8.

Since the lines are parallel and have different y-intercepts, they will never intersect. Therefore, there is no solution to this system because parallel lines do not meet.

Given this analysis, the correct response is:

The statement is true, because the lines are parallel.

...hi.... · Answer

Use the image to answer the question. Does the graph show the system of equations x+y=2 and −x+y=7 ? Should the lines for the system be parallel? (1 point) Responses The graph of x+y=2 is incorrect. The lines should intersect. The graph of x plus y equals 2 is incorrect. The lines should intersect. The graph of −x+y=7 is incorrect. The lines should intersect. The graph of negative x plus y equals 7 is incorrect. The lines should intersect. The graph of −x+y=7 is incorrect. The lines should be parallel. The graph of negative x plus y equals 7 is incorrect. The lines should be parallel. Both graphs are correct. The lines should be parallel. Both graphs are correct. The lines should be parallel. Skip to navigation

GPT-4o mini · Answer

To determine whether the graph correctly represents the system of equations $x + y = 2$ and $-x + y = 7$, we need to analyze the equations to see if they should be parallel or if they intersect.

Rewrite the equations in slope-intercept form (y = mx + b):
- For $x + y = 2$: \[ y = -x + 2 \] The slope is $-1$ and the y-intercept is $2$.
- For $-x + y = 7$: \[ y = x + 7 \] The slope is $1$ and the y-intercept is $7$.
Compare the slopes:
- The first line has a slope of $-1$.
- The second line has a slope of $1$.

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

Thus, the correct conclusion is that the graphs of these equations should intersect. Therefore, if the graph shows them as parallel lines, then it is incorrect.

The best response given this analysis is:

The graph of $−x + y = 7$ is incorrect. The lines should intersect.