Question
Write an equation of the line that passes through the points in the table
x= 0, 1, 2, 3
y= 10, 7, 4, 1
x= 0, 1, 2, 3
y= 10, 7, 4, 1
Answers
GPT-4o mini
To find the equation of the line that passes through the given points, we can start by determining the slope (m) of the line using the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
We can select two points from the table. Let's use the points \((0, 10)\) and \((1, 7)\).
Using these points:
- \(x_1 = 0\), \(y_1 = 10\)
- \(x_2 = 1\), \(y_2 = 7\)
Now, plug in the values:
\[
m = \frac{7 - 10}{1 - 0} = \frac{-3}{1} = -3
\]
Next, we can use the point-slope form of the equation of a line, which is:
\[
y - y_1 = m(x - x_1)
\]
Using the point \((0, 10)\) and the slope \(m = -3\):
\[
y - 10 = -3(x - 0)
\]
\[
y - 10 = -3x
\]
Now, add 10 to both sides to convert it to the slope-intercept form (y = mx + b):
\[
y = -3x + 10
\]
Thus, the equation of the line that passes through the points given in the table is:
\[
\boxed{y = -3x + 10}
\]
You can verify that this equation fits all the points in the table:
1. For \(x = 0\): \(y = -3(0) + 10 = 10\)
2. For \(x = 1\): \(y = -3(1) + 10 = 7\)
3. For \(x = 2\): \(y = -3(2) + 10 = 4\)
4. For \(x = 3\): \(y = -3(3) + 10 = 1\)
All points match, confirming that the equation is correct!
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
We can select two points from the table. Let's use the points \((0, 10)\) and \((1, 7)\).
Using these points:
- \(x_1 = 0\), \(y_1 = 10\)
- \(x_2 = 1\), \(y_2 = 7\)
Now, plug in the values:
\[
m = \frac{7 - 10}{1 - 0} = \frac{-3}{1} = -3
\]
Next, we can use the point-slope form of the equation of a line, which is:
\[
y - y_1 = m(x - x_1)
\]
Using the point \((0, 10)\) and the slope \(m = -3\):
\[
y - 10 = -3(x - 0)
\]
\[
y - 10 = -3x
\]
Now, add 10 to both sides to convert it to the slope-intercept form (y = mx + b):
\[
y = -3x + 10
\]
Thus, the equation of the line that passes through the points given in the table is:
\[
\boxed{y = -3x + 10}
\]
You can verify that this equation fits all the points in the table:
1. For \(x = 0\): \(y = -3(0) + 10 = 10\)
2. For \(x = 1\): \(y = -3(1) + 10 = 7\)
3. For \(x = 2\): \(y = -3(2) + 10 = 4\)
4. For \(x = 3\): \(y = -3(3) + 10 = 1\)
All points match, confirming that the equation is correct!
unf0rgettxble
Use the slope intercept formula
GPT-4o mini
Sure! The slope-intercept form of the equation of a line is given by:
\[
y = mx + b
\]
where \(m\) is the slope and \(b\) is the y-intercept.
### Step 1: Calculate the Slope
We can calculate the slope \(m\) using the two points, for example, \((0, 10)\) and \((1, 7)\):
\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{7 - 10}{1 - 0} = \frac{-3}{1} = -3
\]
### Step 2: Find the Y-Intercept
The y-intercept \(b\) can be found by using one of the points and the slope we just calculated. Let's use the point \((0, 10)\):
From the slope-intercept formula:
\[
y = mx + b
\]
Plugging in \(x = 0\) and \(y = 10\):
\[
10 = -3(0) + b
\]
\[
10 = b
\]
### Step 3: Write the Equation
Now that we know \(m = -3\) and \(b = 10\), we can write the equation of the line in slope-intercept form:
\[
y = -3x + 10
\]
Thus, the equation of the line in slope-intercept form is:
\[
\boxed{y = -3x + 10}
\]
This confirms the line's equation derived previously!
\[
y = mx + b
\]
where \(m\) is the slope and \(b\) is the y-intercept.
### Step 1: Calculate the Slope
We can calculate the slope \(m\) using the two points, for example, \((0, 10)\) and \((1, 7)\):
\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{7 - 10}{1 - 0} = \frac{-3}{1} = -3
\]
### Step 2: Find the Y-Intercept
The y-intercept \(b\) can be found by using one of the points and the slope we just calculated. Let's use the point \((0, 10)\):
From the slope-intercept formula:
\[
y = mx + b
\]
Plugging in \(x = 0\) and \(y = 10\):
\[
10 = -3(0) + b
\]
\[
10 = b
\]
### Step 3: Write the Equation
Now that we know \(m = -3\) and \(b = 10\), we can write the equation of the line in slope-intercept form:
\[
y = -3x + 10
\]
Thus, the equation of the line in slope-intercept form is:
\[
\boxed{y = -3x + 10}
\]
This confirms the line's equation derived previously!
unf0rgettxble
Now do this table
x= -2, 0, 2, 3
y= -17, -5, 7, 13
x= -2, 0, 2, 3
y= -17, -5, 7, 13
GPT-4o mini
To find the equation of the line that passes through the given points, we will again use the slope-intercept form of the equation of a line defined by:
\[
y = mx + b
\]
### Step 1: Calculate the Slope
Let's select two points from the table. We can use the points \((-2, -17)\) and \((0, -5)\) to calculate the slope \(m\):
Using the formula for slope:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Let:
- \(x_1 = -2\), \(y_1 = -17\)
- \(x_2 = 0\), \(y_2 = -5\)
Now plug in the values:
\[
m = \frac{-5 - (-17)}{0 - (-2)} = \frac{-5 + 17}{0 + 2} = \frac{12}{2} = 6
\]
### Step 2: Find the Y-Intercept
Now we need to find the y-intercept \(b\). We can use one of the points along with the slope \(m\). Let’s use the point \((0, -5)\):
From the slope-intercept formula:
\[
y = mx + b
\]
Plugging in \(x = 0\) and \(y = -5\):
\[
-5 = 6(0) + b
\]
\[
-5 = b
\]
### Step 3: Write the Equation
Now that we have found \(m = 6\) and \(b = -5\), we can write the equation of the line in slope-intercept form:
\[
y = 6x - 5
\]
### Conclusion
Thus, the equation of the line that passes through the points in the table is:
\[
\boxed{y = 6x - 5}
\]
### Verification:
We can verify that this equation fits all the given points:
1. **For \(x = -2\)**:
\[
y = 6(-2) - 5 = -12 - 5 = -17
\]
2. **For \(x = 0\)**:
\[
y = 6(0) - 5 = 0 - 5 = -5
\]
3. **For \(x = 2\)**:
\[
y = 6(2) - 5 = 12 - 5 = 7
\]
4. **For \(x = 3\)**:
\[
y = 6(3) - 5 = 18 - 5 = 13
\]
All points match the original table, confirming that the equation is correct!
\[
y = mx + b
\]
### Step 1: Calculate the Slope
Let's select two points from the table. We can use the points \((-2, -17)\) and \((0, -5)\) to calculate the slope \(m\):
Using the formula for slope:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Let:
- \(x_1 = -2\), \(y_1 = -17\)
- \(x_2 = 0\), \(y_2 = -5\)
Now plug in the values:
\[
m = \frac{-5 - (-17)}{0 - (-2)} = \frac{-5 + 17}{0 + 2} = \frac{12}{2} = 6
\]
### Step 2: Find the Y-Intercept
Now we need to find the y-intercept \(b\). We can use one of the points along with the slope \(m\). Let’s use the point \((0, -5)\):
From the slope-intercept formula:
\[
y = mx + b
\]
Plugging in \(x = 0\) and \(y = -5\):
\[
-5 = 6(0) + b
\]
\[
-5 = b
\]
### Step 3: Write the Equation
Now that we have found \(m = 6\) and \(b = -5\), we can write the equation of the line in slope-intercept form:
\[
y = 6x - 5
\]
### Conclusion
Thus, the equation of the line that passes through the points in the table is:
\[
\boxed{y = 6x - 5}
\]
### Verification:
We can verify that this equation fits all the given points:
1. **For \(x = -2\)**:
\[
y = 6(-2) - 5 = -12 - 5 = -17
\]
2. **For \(x = 0\)**:
\[
y = 6(0) - 5 = 0 - 5 = -5
\]
3. **For \(x = 2\)**:
\[
y = 6(2) - 5 = 12 - 5 = 7
\]
4. **For \(x = 3\)**:
\[
y = 6(3) - 5 = 18 - 5 = 13
\]
All points match the original table, confirming that the equation is correct!
unf0rgettxble
I'm not saying your wrong, I'm just wondering why the plug in was -5, and not -17?
GPT-4o mini
The choice of which point to use for finding the y-intercept \(b\) in the slope-intercept form \(y = mx + b\) is somewhat arbitrary as long as you use a point where you know both the \(x\) and \(y\) values.
In general, you would want to select the point where \(x = 0\) because that directly gives you the y-intercept \(b\). For the equation format \(y = mx + b\):
- When \(x = 0\), then \(y = b\). This directly tells you the value of \(b\).
However, it’s perfectly fine to use any point to solve for \(b\). If you used the point \((-2, -17)\) instead, you would proceed as follows:
1. With \(m = 6\) (the slope we calculated earlier) and using the point \((-2, -17)\):
\[
y = mx + b
\]
Substituting the values:
\[
-17 = 6(-2) + b
\]
\[
-17 = -12 + b
\]
Now add 12 to both sides:
\[
b = -17 + 12 = -5
\]
This calculation confirms that \(b = -5\), so whether you choose \((0, -5)\) or \((-2, -17)\), you should arrive at the same value for \(b\).
Thus, using either point is valid; the most straightforward method, when possible, is to use the point with \(x = 0\) to find \(b\) directly.
In summary, both approaches yield the same result, and using \(-5\) or \(-17\) simply reflects that you can choose any point to derive the y-intercept. Thank you for asking for clarification!
In general, you would want to select the point where \(x = 0\) because that directly gives you the y-intercept \(b\). For the equation format \(y = mx + b\):
- When \(x = 0\), then \(y = b\). This directly tells you the value of \(b\).
However, it’s perfectly fine to use any point to solve for \(b\). If you used the point \((-2, -17)\) instead, you would proceed as follows:
1. With \(m = 6\) (the slope we calculated earlier) and using the point \((-2, -17)\):
\[
y = mx + b
\]
Substituting the values:
\[
-17 = 6(-2) + b
\]
\[
-17 = -12 + b
\]
Now add 12 to both sides:
\[
b = -17 + 12 = -5
\]
This calculation confirms that \(b = -5\), so whether you choose \((0, -5)\) or \((-2, -17)\), you should arrive at the same value for \(b\).
Thus, using either point is valid; the most straightforward method, when possible, is to use the point with \(x = 0\) to find \(b\) directly.
In summary, both approaches yield the same result, and using \(-5\) or \(-17\) simply reflects that you can choose any point to derive the y-intercept. Thank you for asking for clarification!
unf0rgettxble
Could you simplify everything you just said
GPT-4o mini
Certainly! Here’s a simplified explanation:
1. **Choosing Points**: You can use any point on the line to find the y-intercept \(b\), but it's easiest to use the point where \(x = 0\) because directly gives you \(b\).
2. **Using \((0, -5)\)**:
- Since \(x = 0\), \(b = -5\) is directly available.
3. **Using \((-2, -17)\)**:
- If you use this point, you still find \(b = -5\) by plugging it into the equation:
\[
-17 = 6(-2) + b \implies -17 = -12 + b \implies b = -5
\]
So, whether you use \((0, -5)\) or \((-2, -17)\), you arrive at the same y-intercept: \(b = -5\). It's just easier to use the point where \(x=0\) for finding \(b\).
I hope this clarifies everything!
1. **Choosing Points**: You can use any point on the line to find the y-intercept \(b\), but it's easiest to use the point where \(x = 0\) because directly gives you \(b\).
2. **Using \((0, -5)\)**:
- Since \(x = 0\), \(b = -5\) is directly available.
3. **Using \((-2, -17)\)**:
- If you use this point, you still find \(b = -5\) by plugging it into the equation:
\[
-17 = 6(-2) + b \implies -17 = -12 + b \implies b = -5
\]
So, whether you use \((0, -5)\) or \((-2, -17)\), you arrive at the same y-intercept: \(b = -5\). It's just easier to use the point where \(x=0\) for finding \(b\).
I hope this clarifies everything!
unf0rgettxble
Thank you so much!!
Now do this table
x=-6,-4,-2,0,2
y=0,-8,-16,-24,-32
Now do this table
x=-6,-4,-2,0,2
y=0,-8,-16,-24,-32
GPT-4o mini
To find the equation of the line that passes through the points in the provided table, we'll follow the same steps as before, using the slope-intercept form of a line:
\[
y = mx + b
\]
### Step 1: Calculate the Slope
First, we need to calculate the slope \(m\). Let's use the points \((-6, 0)\) and \((-4, -8)\):
Using the formula for slope:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Let:
- \(x_1 = -6\), \(y_1 = 0\)
- \(x_2 = -4\), \(y_2 = -8\)
Now plug in the values:
\[
m = \frac{-8 - 0}{-4 - (-6)} = \frac{-8}{-4 + 6} = \frac{-8}{2} = -4
\]
### Step 2: Find the Y-Intercept
Now that we have the slope \(m = -4\), we will find the y-intercept \(b\). We can use the point where \(x = 0\) if available, but since we do not have it directly, let's use the point \((0, -24)\):
Using the slope-intercept form:
\[
y = mx + b
\]
We can use any point, but we’ll use \((0, -24)\):
\[
-24 = -4(0) + b
\]
\[
-24 = b
\]
### Step 3: Write the Equation
Now that we have \(m = -4\) and \(b = -24\), we can write the equation of the line:
\[
y = -4x - 24
\]
### Verification
Let's verify that this equation fits all the points provided in the table:
1. **For \(x = -6\)**:
\[
y = -4(-6) - 24 = 24 - 24 = 0
\]
2. **For \(x = -4\)**:
\[
y = -4(-4) - 24 = 16 - 24 = -8
\]
3. **For \(x = -2\)**:
\[
y = -4(-2) - 24 = 8 - 24 = -16
\]
4. **For \(x = 0\)**:
\[
y = -4(0) - 24 = 0 - 24 = -24
\]
5. **For \(x = 2\)**:
\[
y = -4(2) - 24 = -8 - 24 = -32
\]
All points match, confirming that the equation is correct!
### Final Answer
The equation of the line that passes through the points given in the table is:
\[
\boxed{y = -4x - 24}
\]
\[
y = mx + b
\]
### Step 1: Calculate the Slope
First, we need to calculate the slope \(m\). Let's use the points \((-6, 0)\) and \((-4, -8)\):
Using the formula for slope:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Let:
- \(x_1 = -6\), \(y_1 = 0\)
- \(x_2 = -4\), \(y_2 = -8\)
Now plug in the values:
\[
m = \frac{-8 - 0}{-4 - (-6)} = \frac{-8}{-4 + 6} = \frac{-8}{2} = -4
\]
### Step 2: Find the Y-Intercept
Now that we have the slope \(m = -4\), we will find the y-intercept \(b\). We can use the point where \(x = 0\) if available, but since we do not have it directly, let's use the point \((0, -24)\):
Using the slope-intercept form:
\[
y = mx + b
\]
We can use any point, but we’ll use \((0, -24)\):
\[
-24 = -4(0) + b
\]
\[
-24 = b
\]
### Step 3: Write the Equation
Now that we have \(m = -4\) and \(b = -24\), we can write the equation of the line:
\[
y = -4x - 24
\]
### Verification
Let's verify that this equation fits all the points provided in the table:
1. **For \(x = -6\)**:
\[
y = -4(-6) - 24 = 24 - 24 = 0
\]
2. **For \(x = -4\)**:
\[
y = -4(-4) - 24 = 16 - 24 = -8
\]
3. **For \(x = -2\)**:
\[
y = -4(-2) - 24 = 8 - 24 = -16
\]
4. **For \(x = 0\)**:
\[
y = -4(0) - 24 = 0 - 24 = -24
\]
5. **For \(x = 2\)**:
\[
y = -4(2) - 24 = -8 - 24 = -32
\]
All points match, confirming that the equation is correct!
### Final Answer
The equation of the line that passes through the points given in the table is:
\[
\boxed{y = -4x - 24}
\]
unf0rgettxble
Now please do this table
x= -2,-1.5,-1,-0.5,0,0.5,1
y= -0.4,1,2.4,3.8,5.2,6.6,8
x= -2,-1.5,-1,-0.5,0,0.5,1
y= -0.4,1,2.4,3.8,5.2,6.6,8
GPT-4o mini
To find the equation of the line that passes through the points in the provided table, we'll follow the same steps as before, using the slope-intercept form:
\[
y = mx + b
\]
### Step 1: Calculate the Slope
Let’s use two points from the table, say \((-2, -0.4)\) and \((0.5, 6.6)\), to calculate the slope \(m\).
Using the formula for slope:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Let's define:
- \(x_1 = -2\), \(y_1 = -0.4\)
- \(x_2 = 0.5\), \(y_2 = 6.6\)
Now, plug in the values:
\[
m = \frac{6.6 - (-0.4)}{0.5 - (-2)} = \frac{6.6 + 0.4}{0.5 + 2} = \frac{7}{2.5} = \frac{7 \times 2}{2.5 \times 2} = \frac{14}{5} = 2.8
\]
### Step 2: Find the Y-Intercept
Now that we have the slope \(m = 2.8\), we will find the y-intercept \(b\). We can use any point; let’s use the point \((0, 5.2)\):
Using the slope-intercept form:
\[
y = mx + b
\]
Using the point \((0, 5.2)\):
\[
5.2 = 2.8(0) + b
\]
\[
5.2 = b
\]
### Step 3: Write the Equation
Now that we have \(m = 2.8\) and \(b = 5.2\), we can write the equation of the line:
\[
y = 2.8x + 5.2
\]
### Verification
Let's check if this equation fits all the points provided in the table:
1. **For \(x = -2\)**:
\[
y = 2.8(-2) + 5.2 = -5.6 + 5.2 = -0.4
\]
2. **For \(x = -1.5\)**:
\[
y = 2.8(-1.5) + 5.2 = -4.2 + 5.2 = 1
\]
3. **For \(x = -1\)**:
\[
y = 2.8(-1) + 5.2 = -2.8 + 5.2 = 2.4
\]
4. **For \(x = -0.5\)**:
\[
y = 2.8(-0.5) + 5.2 = -1.4 + 5.2 = 3.8
\]
5. **For \(x = 0\)**:
\[
y = 2.8(0) + 5.2 = 0 + 5.2 = 5.2
\]
6. **For \(x = 0.5\)**:
\[
y = 2.8(0.5) + 5.2 = 1.4 + 5.2 = 6.6
\]
7. **For \(x = 1\)**:
\[
y = 2.8(1) + 5.2 = 2.8 + 5.2 = 8
\]
All points match, confirming that the equation is correct!
### Final Answer
The equation of the line that passes through the points given in the table is:
\[
\boxed{y = 2.8x + 5.2}
\]
\[
y = mx + b
\]
### Step 1: Calculate the Slope
Let’s use two points from the table, say \((-2, -0.4)\) and \((0.5, 6.6)\), to calculate the slope \(m\).
Using the formula for slope:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Let's define:
- \(x_1 = -2\), \(y_1 = -0.4\)
- \(x_2 = 0.5\), \(y_2 = 6.6\)
Now, plug in the values:
\[
m = \frac{6.6 - (-0.4)}{0.5 - (-2)} = \frac{6.6 + 0.4}{0.5 + 2} = \frac{7}{2.5} = \frac{7 \times 2}{2.5 \times 2} = \frac{14}{5} = 2.8
\]
### Step 2: Find the Y-Intercept
Now that we have the slope \(m = 2.8\), we will find the y-intercept \(b\). We can use any point; let’s use the point \((0, 5.2)\):
Using the slope-intercept form:
\[
y = mx + b
\]
Using the point \((0, 5.2)\):
\[
5.2 = 2.8(0) + b
\]
\[
5.2 = b
\]
### Step 3: Write the Equation
Now that we have \(m = 2.8\) and \(b = 5.2\), we can write the equation of the line:
\[
y = 2.8x + 5.2
\]
### Verification
Let's check if this equation fits all the points provided in the table:
1. **For \(x = -2\)**:
\[
y = 2.8(-2) + 5.2 = -5.6 + 5.2 = -0.4
\]
2. **For \(x = -1.5\)**:
\[
y = 2.8(-1.5) + 5.2 = -4.2 + 5.2 = 1
\]
3. **For \(x = -1\)**:
\[
y = 2.8(-1) + 5.2 = -2.8 + 5.2 = 2.4
\]
4. **For \(x = -0.5\)**:
\[
y = 2.8(-0.5) + 5.2 = -1.4 + 5.2 = 3.8
\]
5. **For \(x = 0\)**:
\[
y = 2.8(0) + 5.2 = 0 + 5.2 = 5.2
\]
6. **For \(x = 0.5\)**:
\[
y = 2.8(0.5) + 5.2 = 1.4 + 5.2 = 6.6
\]
7. **For \(x = 1\)**:
\[
y = 2.8(1) + 5.2 = 2.8 + 5.2 = 8
\]
All points match, confirming that the equation is correct!
### Final Answer
The equation of the line that passes through the points given in the table is:
\[
\boxed{y = 2.8x + 5.2}
\]