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

Answer 1

Use the slope intercept formula

Answer 2

Now do this table

x= -2, 0, 2, 3
y= -17, -5, 7, 13

Answer 3

I'm not saying your wrong, I'm just wondering why the plug in was -5, and not -17?

Answer 4

Could you simplify everything you just said

Answer 5

Thank you so much!!
Now do this table

x=-6,-4,-2,0,2
y=0,-8,-16,-24,-32

Answer 6

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

Answer 7

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!

Answer 8

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!

Answer 9

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!

Answer 10

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!

Answer 11

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!

Answer 12

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:

\[
\boxed{y = -4x - 24}
\]

Answer 13

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:

\[
\boxed{y = 2.8x + 5.2}
\]