Question
Use the image to answer the question.
y = -x + 5
2x + y = 15
Find the coordinates of the intersection point that solves the system of these two linear equations in two variables: 2x + y = 15 and y = -x + 5.
A. (-5, 10)
B. (-10, 5)
C. (10, -5)
D. (5, -10)
y = -x + 5
2x + y = 15
Find the coordinates of the intersection point that solves the system of these two linear equations in two variables: 2x + y = 15 and y = -x + 5.
A. (-5, 10)
B. (-10, 5)
C. (10, -5)
D. (5, -10)
Answers
Answer
Is (-2, 6) a solution to the system of these linear equations: x + 2y = 10 and 3x + y = 0? Why?
a. No, because the graphs intersect at (-2, 6)
b. Yes, because the graphs don't intersect at (-2, 6)
c. No, because the graphs don't intersect at (-2, 6).
d. Yes, because the graphs intersect at (-2, 6).
a. No, because the graphs intersect at (-2, 6)
b. Yes, because the graphs don't intersect at (-2, 6)
c. No, because the graphs don't intersect at (-2, 6).
d. Yes, because the graphs intersect at (-2, 6).
Answer
Use the image to answer the question.
y = -2x + 3
y = 5x + 10
Estimate the solution to the system of these linear equations based on the graph.
y = -2x + 3
y = 5x + 10
a. (-1, 5)
b. (5, -1)
c. (1, 5)
d. (-1, -5)
y = -2x + 3
y = 5x + 10
Estimate the solution to the system of these linear equations based on the graph.
y = -2x + 3
y = 5x + 10
a. (-1, 5)
b. (5, -1)
c. (1, 5)
d. (-1, -5)
Answer
What is the last step to solving a system of equations?
a. check the answer.
b. estimate the intersection point.
c. make a table of solutions of the linear equations.
d. graph the lines.
a. check the answer.
b. estimate the intersection point.
c. make a table of solutions of the linear equations.
d. graph the lines.
Answer
Use the image to answer the question.
y = -5x + 7
y = 2x + 10
Estimate the solution to the system of these linear equations based on the graph.
y = -5x + 7
y = 2x + 10
a. (12, -1)
b. (1, -12)
c. (-1, 12)
d. (-1, -12)
y = -5x + 7
y = 2x + 10
Estimate the solution to the system of these linear equations based on the graph.
y = -5x + 7
y = 2x + 10
a. (12, -1)
b. (1, -12)
c. (-1, 12)
d. (-1, -12)
Answer
Correct answers
1.(10, -5)
2.Yes, because the graphs intersect at (-2, 6).
3. (-1, 5)
4. check the answer.
5. (-1, 12)
1.(10, -5)
2.Yes, because the graphs intersect at (-2, 6).
3. (-1, 5)
4. check the answer.
5. (-1, 12)
Answered by
GPT-4o mini
To find the intersection point of the two equations given, we can substitute the expression for \(y\) from the first equation into the second equation.
1. Start with the equations:
- \(y = -x + 5\)
- \(2x + y = 15\)
2. Substitute \(y\) from the first equation into the second equation:
\[
2x + (-x + 5) = 15
\]
Simplify:
\[
2x - x + 5 = 15
\]
\[
x + 5 = 15
\]
\[
x = 15 - 5
\]
\[
x = 10
\]
3. Now substitute \(x = 10\) back into the first equation to find \(y\):
\[
y = -10 + 5
\]
\[
y = -5
\]
Thus, the coordinates of the intersection point are \((10, -5)\). Therefore, the correct answer is:
C. (10, -5)
1. Start with the equations:
- \(y = -x + 5\)
- \(2x + y = 15\)
2. Substitute \(y\) from the first equation into the second equation:
\[
2x + (-x + 5) = 15
\]
Simplify:
\[
2x - x + 5 = 15
\]
\[
x + 5 = 15
\]
\[
x = 15 - 5
\]
\[
x = 10
\]
3. Now substitute \(x = 10\) back into the first equation to find \(y\):
\[
y = -10 + 5
\]
\[
y = -5
\]
Thus, the coordinates of the intersection point are \((10, -5)\). Therefore, the correct answer is:
C. (10, -5)
Answered by
GPT-4o mini
To determine whether \((-2, 6)\) is a solution to the system of equations given by \(x + 2y = 10\) and \(3x + y = 0\), we will substitute \(-2\) for \(x\) and \(6\) for \(y\) in both equations.
1. For the first equation \(x + 2y = 10\):
\[
-2 + 2(6) = 10
\]
\[
-2 + 12 = 10
\]
\[
10 = 10 \quad \text{(True)}
\]
2. For the second equation \(3x + y = 0\):
\[
3(-2) + 6 = 0
\]
\[
-6 + 6 = 0
\]
\[
0 = 0 \quad \text{(True)}
\]
Since \((-2, 6)\) satisfies both equations, it is indeed a solution to the system.
Now, let's analyze the options:
- a. No, because the graphs intersect at (-2, 6)
- b. Yes, because the graphs don't intersect at (-2, 6)
- c. No, because the graphs don't intersect at (-2, 6).
- d. Yes, because the graphs intersect at (-2, 6).
Since we have confirmed that \((-2, 6)\) is a solution and the graphs do intersect at that point, the correct answer is:
**d. Yes, because the graphs intersect at (-2, 6).**
1. For the first equation \(x + 2y = 10\):
\[
-2 + 2(6) = 10
\]
\[
-2 + 12 = 10
\]
\[
10 = 10 \quad \text{(True)}
\]
2. For the second equation \(3x + y = 0\):
\[
3(-2) + 6 = 0
\]
\[
-6 + 6 = 0
\]
\[
0 = 0 \quad \text{(True)}
\]
Since \((-2, 6)\) satisfies both equations, it is indeed a solution to the system.
Now, let's analyze the options:
- a. No, because the graphs intersect at (-2, 6)
- b. Yes, because the graphs don't intersect at (-2, 6)
- c. No, because the graphs don't intersect at (-2, 6).
- d. Yes, because the graphs intersect at (-2, 6).
Since we have confirmed that \((-2, 6)\) is a solution and the graphs do intersect at that point, the correct answer is:
**d. Yes, because the graphs intersect at (-2, 6).**
Answered by
GPT-4o mini
To determine the estimated solution to the system of equations \(y = -2x + 3\) and \(y = 5x + 10\) based on a graph, we can analyze the lines represented by these equations:
1. **Equation 1: \(y = -2x + 3\)**
- The slope is \(-2\), which means it goes down 2 units for every 1 unit it moves to the right.
- The y-intercept is \(3\) (the point where the line crosses the y-axis).
2. **Equation 2: \(y = 5x + 10\)**
- The slope is \(5\), which means it goes up 5 units for every 1 unit it moves to the right.
- The y-intercept is \(10\) (the point where the line crosses the y-axis).
### Finding the Intersection Point
To estimate the intersection point, we can look at the values provided:
a. \((-1, 5)\)
b. \((5, -1)\)
c. \((1, 5)\)
d. \((-1, -5)\)
Let's check each option to see if they satisfy both equations, thus verifying if any of those points is the intersection.
#### a. (-1, 5)
- For \(y = -2x + 3\):
\[
5 = -2(-1) + 3 \implies 5 = 2 + 3 \implies 5 = 5 \quad \text{(True)}
\]
- For \(y = 5x + 10\):
\[
5 = 5(-1) + 10 \implies 5 = -5 + 10 \implies 5 = 5 \quad \text{(True)}
\]
- So, \((-1, 5)\) is a solution.
#### b. (5, -1)
- For \(y = -2x + 3\):
\[
-1 = -2(5) + 3 \implies -1 = -10 + 3 \implies -1 = -7 \quad \text{(False)}
\]
- Therefore, (5, -1) is not a solution.
#### c. (1, 5)
- For \(y = -2x + 3\):
\[
5 = -2(1) + 3 \implies 5 = -2 + 3 \implies 5 = 1 \quad \text{(False)}
\]
- Thus, (1, 5) is not a solution.
#### d. (-1, -5)
- For \(y = -2x + 3\):
\[
-5 = -2(-1) + 3 \implies -5 = 2 + 3 \implies -5 = 5 \quad \text{(False)}
\]
- Therefore, (-1, -5) is not a solution.
### Conclusion
The point \((-1, 5)\) satisfies both equations, making it the intersection point based on your options. Therefore, the correct answer is:
**a. (-1, 5)**
1. **Equation 1: \(y = -2x + 3\)**
- The slope is \(-2\), which means it goes down 2 units for every 1 unit it moves to the right.
- The y-intercept is \(3\) (the point where the line crosses the y-axis).
2. **Equation 2: \(y = 5x + 10\)**
- The slope is \(5\), which means it goes up 5 units for every 1 unit it moves to the right.
- The y-intercept is \(10\) (the point where the line crosses the y-axis).
### Finding the Intersection Point
To estimate the intersection point, we can look at the values provided:
a. \((-1, 5)\)
b. \((5, -1)\)
c. \((1, 5)\)
d. \((-1, -5)\)
Let's check each option to see if they satisfy both equations, thus verifying if any of those points is the intersection.
#### a. (-1, 5)
- For \(y = -2x + 3\):
\[
5 = -2(-1) + 3 \implies 5 = 2 + 3 \implies 5 = 5 \quad \text{(True)}
\]
- For \(y = 5x + 10\):
\[
5 = 5(-1) + 10 \implies 5 = -5 + 10 \implies 5 = 5 \quad \text{(True)}
\]
- So, \((-1, 5)\) is a solution.
#### b. (5, -1)
- For \(y = -2x + 3\):
\[
-1 = -2(5) + 3 \implies -1 = -10 + 3 \implies -1 = -7 \quad \text{(False)}
\]
- Therefore, (5, -1) is not a solution.
#### c. (1, 5)
- For \(y = -2x + 3\):
\[
5 = -2(1) + 3 \implies 5 = -2 + 3 \implies 5 = 1 \quad \text{(False)}
\]
- Thus, (1, 5) is not a solution.
#### d. (-1, -5)
- For \(y = -2x + 3\):
\[
-5 = -2(-1) + 3 \implies -5 = 2 + 3 \implies -5 = 5 \quad \text{(False)}
\]
- Therefore, (-1, -5) is not a solution.
### Conclusion
The point \((-1, 5)\) satisfies both equations, making it the intersection point based on your options. Therefore, the correct answer is:
**a. (-1, 5)**
Answered by
GPT-4o mini
The last step to solving a system of equations is generally to verify that your solution is correct. This is often done by substituting the solution back into the original equations to ensure both equations hold true.
Therefore, the correct answer is:
**a. check the answer.**
Therefore, the correct answer is:
**a. check the answer.**
Answered by
GPT-4o mini
To estimate the solution to the system of equations \(y = -5x + 7\) and \(y = 2x + 10\) based on a graph, we would typically look for the intersection point of the two lines represented by these equations.
Here’s a breakdown of what each option represents in terms of potential intersection points:
### Checking Each Option
**a. (12, -1)**
- For \(y = -5x + 7\):
\[
-1 = -5(12) + 7 \implies -1 = -60 + 7 \implies -1 = -53 \quad \text{(False)}
\]
**b. (1, -12)**
- For \(y = -5x + 7\):
\[
-12 = -5(1) + 7 \implies -12 = -5 + 7 \implies -12 = 2 \quad \text{(False)}
\]
**c. (-1, 12)**
- For \(y = -5x + 7\):
\[
12 = -5(-1) + 7 \implies 12 = 5 + 7 \implies 12 = 12 \quad \text{(True)}
\]
- For \(y = 2x + 10\):
\[
12 = 2(-1) + 10 \implies 12 = -2 + 10 \implies 12 = 8 \quad \text{(False)}
\]
**d. (-1, -12)**
- For \(y = -5x + 7\):
\[
-12 = -5(-1) + 7 \implies -12 = 5 + 7 \implies -12 = 12 \quad \text{(False)}
\]
### Conclusion
From checking each point, \((-1, 12)\) satisfies the first equation but not the second. However, if we were to estimate based on the graph, it seems that none of the given options appear to be intersection points.
If you have the graph displayed, it is possible that the intersection point lies outside the options provided, or there could be an error in approximating the intersection visually.
That said, based solely on substitution checks, it appears the valid candidates do not converge as a solution based on the calculations provided. If the solutions are mandated strictly from the options given, further clarification on the graph may be required, as physical checking against the plotted data would be essential in verification.
If you need a more definitive conclusion or insights based on provided graph visuals, please provide those details so I can tailor the estimation accurately!
Here’s a breakdown of what each option represents in terms of potential intersection points:
### Checking Each Option
**a. (12, -1)**
- For \(y = -5x + 7\):
\[
-1 = -5(12) + 7 \implies -1 = -60 + 7 \implies -1 = -53 \quad \text{(False)}
\]
**b. (1, -12)**
- For \(y = -5x + 7\):
\[
-12 = -5(1) + 7 \implies -12 = -5 + 7 \implies -12 = 2 \quad \text{(False)}
\]
**c. (-1, 12)**
- For \(y = -5x + 7\):
\[
12 = -5(-1) + 7 \implies 12 = 5 + 7 \implies 12 = 12 \quad \text{(True)}
\]
- For \(y = 2x + 10\):
\[
12 = 2(-1) + 10 \implies 12 = -2 + 10 \implies 12 = 8 \quad \text{(False)}
\]
**d. (-1, -12)**
- For \(y = -5x + 7\):
\[
-12 = -5(-1) + 7 \implies -12 = 5 + 7 \implies -12 = 12 \quad \text{(False)}
\]
### Conclusion
From checking each point, \((-1, 12)\) satisfies the first equation but not the second. However, if we were to estimate based on the graph, it seems that none of the given options appear to be intersection points.
If you have the graph displayed, it is possible that the intersection point lies outside the options provided, or there could be an error in approximating the intersection visually.
That said, based solely on substitution checks, it appears the valid candidates do not converge as a solution based on the calculations provided. If the solutions are mandated strictly from the options given, further clarification on the graph may be required, as physical checking against the plotted data would be essential in verification.
If you need a more definitive conclusion or insights based on provided graph visuals, please provide those details so I can tailor the estimation accurately!
Answered by
GPT-4o mini
Thank you for providing the correct answers. Here's a summary based on your responses:
1. **Intersection Point of the System:**
- **Answer:** (10, -5)
2. **Is (-2, 6) a Solution to the System?**
- **Answer:** Yes, because the graphs intersect at (-2, 6).
3. **Estimate the Solution for the Graph of the Equations:**
- **Answer:** (-1, 5)
4. **Final Step to Solving a System of Equations:**
- **Answer:** Check the answer.
5. **Estimate the Solution for the Given Graph:**
- **Answer:** (-1, 12)
If you have any further questions or need more assistance, feel free to ask!
1. **Intersection Point of the System:**
- **Answer:** (10, -5)
2. **Is (-2, 6) a Solution to the System?**
- **Answer:** Yes, because the graphs intersect at (-2, 6).
3. **Estimate the Solution for the Graph of the Equations:**
- **Answer:** (-1, 5)
4. **Final Step to Solving a System of Equations:**
- **Answer:** Check the answer.
5. **Estimate the Solution for the Given Graph:**
- **Answer:** (-1, 12)
If you have any further questions or need more assistance, feel free to ask!
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