Solved on Jan 17, 2024

Find the slope of the line with the given $(x, y)$ values, then write the equation in point-slope form.
Slope $= \frac{-34 - (-19)}{4 - 1} = -5$ Equation in point-slope form: $y - (-19) = -5(x - 1)$

STEP 1

Assumptions
1. The relationship between $x$ and $y$ is linear.
2. The table of values provides pairs of corresponding $x$ and $y$ values.
3. To find the slope, we can use any two distinct points from the table.
4. Point-slope form of a line is given by $y - y_1 = m(x - x_1)$ , where $m$ is the slope and $(x_1, y_1)$ is a point on the line.

STEP 2

To find the slope $m$ of the line, we use the formula:
$m = \frac{\Delta y}{\Delta x}$
where $\Delta y$ is the change in $y$ and $\Delta x$ is the change in $x$ .

STEP 3

Select any two points from the table to calculate the slope. Let's use the points $(x_1, y_1) = (1, -19)$ and $(x_2, y_2) = (2, -24)$ .

STEP 4

Calculate the change in $y$ ( $\Delta y$ ) and the change in $x$ ( $\Delta x$ ) using the selected points.
$\Delta y = y_2 - y_1$ $\Delta x = x_2 - x_1$

STEP 5

Substitute the $y$ values of the selected points into the $\Delta y$ formula.
$\Delta y = -24 - (-19)$

STEP 6

Calculate the change in $y$ .
$\Delta y = -24 + 19 = -5$

STEP 7

Substitute the $x$ values of the selected points into the $\Delta x$ formula.
$\Delta x = 2 - 1$

STEP 8

Calculate the change in $x$ .
$\Delta x = 2 - 1 = 1$

STEP 9

Now, calculate the slope $m$ using the values of $\Delta y$ and $\Delta x$ .
$m = \frac{\Delta y}{\Delta x}$

STEP 10

Substitute the calculated values into the slope formula.
$m = \frac{-5}{1}$

STEP 11

Calculate the slope.
$m = -5$
The slope of the line is $-5$ .

STEP 12

To write the equation of the line in point-slope form, we use the formula:
$y - y_1 = m(x - x_1)$

STEP 13

Choose a point from the table to use in the point-slope formula. We can use the same point as before, $(x_1, y_1) = (1, -19)$ .

STEP 14

Substitute the slope $m$ and the coordinates of the point $(x_1, y_1)$ into the point-slope formula.
$y - (-19) = -5(x - 1)$

STEP 15

Simplify the equation by distributing the slope and combining like terms if necessary.
$y + 19 = -5x + 5$

STEP 16

Write the final equation of the line in point-slope form.
$y - 19 = -5(x - 1)$
This is the equation of the line in point-slope form.

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Find the slope of the line with the given $(x, y)$ values, then write the equation in point-slope form.
Slope $= \frac{-34 - (-19)}{4 - 1} = -5$ Equation in point-slope form: $y - (-19) = -5(x - 1)$

STEP 1

STEP 2

To find the slope $m$ of the line, we use the formula:
$m = \frac{\Delta y}{\Delta x}$
where $\Delta y$ is the change in $y$ and $\Delta x$ is the change in $x$ .

STEP 3

Select any two points from the table to calculate the slope. Let's use the points $(x_1, y_1) = (1, -19)$ and $(x_2, y_2) = (2, -24)$ .

STEP 4

Calculate the change in $y$ ( $\Delta y$ ) and the change in $x$ ( $\Delta x$ ) using the selected points.
$\Delta y = y_2 - y_1$ $\Delta x = x_2 - x_1$

STEP 5

Substitute the $y$ values of the selected points into the $\Delta y$ formula.
$\Delta y = -24 - (-19)$

STEP 6

Calculate the change in $y$ .
$\Delta y = -24 + 19 = -5$

STEP 7

Substitute the $x$ values of the selected points into the $\Delta x$ formula.
$\Delta x = 2 - 1$

STEP 8

Calculate the change in $x$ .
$\Delta x = 2 - 1 = 1$

STEP 9

Now, calculate the slope $m$ using the values of $\Delta y$ and $\Delta x$ .
$m = \frac{\Delta y}{\Delta x}$

STEP 10

Substitute the calculated values into the slope formula.
$m = \frac{-5}{1}$

STEP 11

Calculate the slope.
$m = -5$
The slope of the line is $-5$ .

STEP 12

To write the equation of the line in point-slope form, we use the formula:
$y - y_1 = m(x - x_1)$

STEP 13

Choose a point from the table to use in the point-slope formula. We can use the same point as before, $(x_1, y_1) = (1, -19)$ .

STEP 14

Substitute the slope $m$ and the coordinates of the point $(x_1, y_1)$ into the point-slope formula.
$y - (-19) = -5(x - 1)$

STEP 15

Simplify the equation by distributing the slope and combining like terms if necessary.
$y + 19 = -5x + 5$

STEP 16

Write the final equation of the line in point-slope form.
$y - 19 = -5(x - 1)$
This is the equation of the line in point-slope form.

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STEP 1

STEP 2

To find the slope mmm of the line, we use the formula:m=ΔyΔxm = \frac{\Delta y}{\Delta x}m=ΔxΔy​where Δy\Delta yΔy is the change in yyy and Δx\Delta xΔx is the change in xxx.

STEP 3

Select any two points from the table to calculate the slope. Let's use the points (x1,y1)=(1,−19)(x_1, y_1) = (1, -19)(x1​,y1​)=(1,−19) and (x2,y2)=(2,−24)(x_2, y_2) = (2, -24)(x2​,y2​)=(2,−24).

STEP 4

Calculate the change in yyy (Δy\Delta yΔy) and the change in xxx (Δx\Delta xΔx) using the selected points.Δy=y2−y1\Delta y = y_2 - y_1Δy=y2​−y1​ Δx=x2−x1\Delta x = x_2 - x_1Δx=x2​−x1​

STEP 5

Substitute the yyy values of the selected points into the Δy\Delta yΔy formula.Δy=−24−(−19)\Delta y = -24 - (-19)Δy=−24−(−19)

STEP 6

Calculate the change in yyy.Δy=−24+19=−5\Delta y = -24 + 19 = -5Δy=−24+19=−5

STEP 7

Substitute the xxx values of the selected points into the Δx\Delta xΔx formula.Δx=2−1\Delta x = 2 - 1Δx=2−1

STEP 8

Calculate the change in xxx.Δx=2−1=1\Delta x = 2 - 1 = 1Δx=2−1=1

STEP 9

Now, calculate the slope mmm using the values of Δy\Delta yΔy and Δx\Delta xΔx.m=ΔyΔxm = \frac{\Delta y}{\Delta x}m=ΔxΔy​

STEP 10

Substitute the calculated values into the slope formula.m=−51m = \frac{-5}{1}m=1−5​

STEP 11

Calculate the slope.m=−5m = -5m=−5The slope of the line is −5-5−5.

STEP 12

To write the equation of the line in point-slope form, we use the formula:y−y1=m(x−x1)y - y_1 = m(x - x_1)y−y1​=m(x−x1​)

STEP 13

Choose a point from the table to use in the point-slope formula. We can use the same point as before, (x1,y1)=(1,−19)(x_1, y_1) = (1, -19)(x1​,y1​)=(1,−19).

STEP 14

Substitute the slope mmm and the coordinates of the point (x1,y1)(x_1, y_1)(x1​,y1​) into the point-slope formula.y−(−19)=−5(x−1)y - (-19) = -5(x - 1)y−(−19)=−5(x−1)

STEP 15

Simplify the equation by distributing the slope and combining like terms if necessary.y+19=−5x+5y + 19 = -5x + 5y+19=−5x+5

STEP 16

Write the final equation of the line in point-slope form.y−19=−5(x−1)y - 19 = -5(x - 1)y−19=−5(x−1)This is the equation of the line in point-slope form.

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To find the slope $m$ of the line, we use the formula:
$m = \frac{\Delta y}{\Delta x}$
where $\Delta y$ is the change in $y$ and $\Delta x$ is the change in $x$ .

Select any two points from the table to calculate the slope. Let's use the points $(x_1, y_1) = (1, -19)$ and $(x_2, y_2) = (2, -24)$ .

Calculate the change in $y$ ( $\Delta y$ ) and the change in $x$ ( $\Delta x$ ) using the selected points.
$\Delta y = y_2 - y_1$ $\Delta x = x_2 - x_1$

Substitute the $y$ values of the selected points into the $\Delta y$ formula.
$\Delta y = -24 - (-19)$

Calculate the change in $y$ .
$\Delta y = -24 + 19 = -5$

Substitute the $x$ values of the selected points into the $\Delta x$ formula.
$\Delta x = 2 - 1$

Calculate the change in $x$ .
$\Delta x = 2 - 1 = 1$

Now, calculate the slope $m$ using the values of $\Delta y$ and $\Delta x$ .
$m = \frac{\Delta y}{\Delta x}$

Substitute the calculated values into the slope formula.
$m = \frac{-5}{1}$

Calculate the slope.
$m = -5$
The slope of the line is $-5$ .

To write the equation of the line in point-slope form, we use the formula:
$y - y_1 = m(x - x_1)$

Choose a point from the table to use in the point-slope formula. We can use the same point as before, $(x_1, y_1) = (1, -19)$ .

Substitute the slope $m$ and the coordinates of the point $(x_1, y_1)$ into the point-slope formula.
$y - (-19) = -5(x - 1)$

Simplify the equation by distributing the slope and combining like terms if necessary.
$y + 19 = -5x + 5$

Write the final equation of the line in point-slope form.
$y - 19 = -5(x - 1)$
This is the equation of the line in point-slope form.