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Write the linear equation in slope-intercept form (use format: no spaces, y=mx+b) of the line passing through the given two points.

(3,1) and (-18,-6)

17 answers

To find the linear equation in slope-intercept form (y = mx + b) for the line passing through the points (3, 1) and (-18, -6), we first need to calculate the slope (m) using the formula:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Here, \((x_1, y_1) = (3, 1)\) and \((x_2, y_2) = (-18, -6)\).

Substituting in the values:

\[ m = \frac{-6 - 1}{-18 - 3} = \frac{-7}{-21} = \frac{1}{3} \]

Now that we have the slope \(m = \frac{1}{3}\), we can use one of the points to find the y-intercept (b). We'll use the point (3, 1):

Using the equation:

\[ y = mx + b \]

Substituting \(x = 3\), \(y = 1\), and \(m = \frac{1}{3}\):

\[ 1 = \frac{1}{3}(3) + b \]

This simplifies to:

\[ 1 = 1 + b \]

To solve for \(b\):

\[ b = 1 - 1 = 0 \]

So, the y-intercept \(b = 0\). Now we can write the slope-intercept form of the equation:

\[ y = \frac{1}{3}x + 0 \]

This simplifies to:

\[ y = \frac{1}{3}x \]

In the required format, the final answer is:

\[ y=\frac{1}{3}x \]

(4,0) and (-14,-9)

To find the linear equation in slope-intercept form (y = mx + b) for the line passing through the points (4, 0) and (-14, -9), we start by calculating the slope (m) using the formula:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Here, \((x_1, y_1) = (4, 0)\) and \((x_2, y_2) = (-14, -9)\).

Substituting in the values:

\[ m = \frac{-9 - 0}{-14 - 4} = \frac{-9}{-18} = \frac{1}{2} \]

Now that we have the slope \(m = \frac{1}{2}\), we need to find the y-intercept (b). We can use the point (4, 0):

Using the equation:

\[ y = mx + b \]

Substituting \(x = 4\), \(y = 0\), and \(m = \frac{1}{2}\):

\[ 0 = \frac{1}{2}(4) + b \]

This simplifies to:

\[ 0 = 2 + b \]

To solve for \(b\):

\[ b = 0 - 2 = -2 \]

So, the y-intercept \(b = -2\). Thus, we can write the slope-intercept form of the equation:

\[ y = \frac{1}{2}x - 2 \]

In the required format, the final answer is:

\[ y=\frac{1}{2}x-2 \]

(-1,-2) and (4,33)

To find the linear equation in slope-intercept form (y = mx + b) for the line passing through the points (-1, -2) and (4, 33), we begin by calculating the slope (m) using the formula:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Here, \((x_1, y_1) = (-1, -2)\) and \((x_2, y_2) = (4, 33)\).

Substituting the values:

\[ m = \frac{33 - (-2)}{4 - (-1)} = \frac{33 + 2}{4 + 1} = \frac{35}{5} = 7 \]

Now that we have the slope \(m = 7\), we can use one of the points to find the y-intercept (b). We'll use the point (-1, -2):

Using the equation:

\[ y = mx + b \]

Substituting \(x = -1\), \(y = -2\), and \(m = 7\):

\[ -2 = 7(-1) + b \]

Solving for \(b\):

\[ -2 = -7 + b \] \[ b = -2 + 7 = 5 \]

Now that we have \(m = 7\) and \(b = 5\), we can write the slope-intercept form of the equation:

\[ y = 7x + 5 \]

In the required format, the final answer is:

\[ y=7x+5 \]

(5,5) and (-5,-15)

To find the linear equation in slope-intercept form (y = mx + b) for the line passing through the points (5, 5) and (-5, -15), we first need to calculate the slope (m) using the formula:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Here, \((x_1, y_1) = (5, 5)\) and \((x_2, y_2) = (-5, -15)\).

Substituting the values:

\[ m = \frac{-15 - 5}{-5 - 5} = \frac{-20}{-10} = 2 \]

Now that we have the slope \(m = 2\), we can use one of the points to find the y-intercept (b). We can use the point (5, 5):

Using the equation:

\[ y = mx + b \]

Substituting \(x = 5\), \(y = 5\), and \(m = 2\):

\[ 5 = 2(5) + b \]

This simplifies to:

\[ 5 = 10 + b \]

To solve for \(b\):

\[ b = 5 - 10 = -5 \]

Now that we have \(m = 2\) and \(b = -5\), we can write the slope-intercept form of the equation:

\[ y = 2x - 5 \]

In the required format, the final answer is:

\[ y=2x-5 \]

(-4,0) and (0,3)

To find the linear equation in slope-intercept form (y = mx + b) for the line passing through the points (-4, 0) and (0, 3), we first need to calculate the slope (m) using the formula:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Here, \((x_1, y_1) = (-4, 0)\) and \((x_2, y_2) = (0, 3)\).

Substituting in the values:

\[ m = \frac{3 - 0}{0 - (-4)} = \frac{3}{4} \]

Now that we have the slope \(m = \frac{3}{4}\), we can use one of the points to find the y-intercept (b). We can use the point (0, 3):

Using the equation:

\[ y = mx + b \]

Substituting \(x = 0\), \(y = 3\), and \(m = \frac{3}{4}\):

\[ 3 = \frac{3}{4}(0) + b \]

This simplifies to:

\[ b = 3 \]

Now that we have \(m = \frac{3}{4}\) and \(b = 3\), we can write the slope-intercept form of the equation:

\[ y = \frac{3}{4}x + 3 \]

In the required format, the final answer is:

\[ y=\frac{3}{4}x+3 \]

(-1,-2) and (-5,-26)

To find the linear equation in slope-intercept form (y = mx + b) for the line passing through the points (-1, -2) and (-5, -26), we first calculate the slope (m) using the formula:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Here, \((x_1, y_1) = (-1, -2)\) and \((x_2, y_2) = (-5, -26)\).

Substituting the values:

\[ m = \frac{-26 - (-2)}{-5 - (-1)} = \frac{-26 + 2}{-5 + 1} = \frac{-24}{-4} = 6 \]

Now that we have the slope \(m = 6\), we can use one of the points to find the y-intercept (b). We'll use the point (-1, -2):

Using the equation:

\[ y = mx + b \]

Substituting \(x = -1\), \(y = -2\), and \(m = 6\):

\[ -2 = 6(-1) + b \]

This simplifies to:

\[ -2 = -6 + b \]

To solve for \(b\):

\[ b = -2 + 6 = 4 \]

Now that we have \(m = 6\) and \(b = 4\), we can write the slope-intercept form of the equation:

\[ y = 6x + 4 \]

In the required format, the final answer is:

\[ y=6x+4 \]

(2,-4) and (8,-1)

To find the linear equation in slope-intercept form (y = mx + b) for the line passing through the points (2, -4) and (8, -1), we will first calculate the slope (m) using the formula:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Here, \((x_1, y_1) = (2, -4)\) and \((x_2, y_2) = (8, -1)\).

Substituting the values:

\[ m = \frac{-1 - (-4)}{8 - 2} = \frac{-1 + 4}{8 - 2} = \frac{3}{6} = \frac{1}{2} \]

Now that we have the slope \(m = \frac{1}{2}\), we can use one of the points to find the y-intercept (b). We will use the point (2, -4):

Using the equation:

\[ y = mx + b \]

Substituting \(x = 2\), \(y = -4\), and \(m = \frac{1}{2}\):

\[ -4 = \frac{1}{2}(2) + b \]

This simplifies to:

\[ -4 = 1 + b \]

To solve for \(b\):

\[ b = -4 - 1 = -5 \]

Now that we have \(m = \frac{1}{2}\) and \(b = -5\), we can write the slope-intercept form of the equation:

\[ y = \frac{1}{2}x - 5 \]

In the required format, the final answer is:

\[ y=\frac{1}{2}x-5 \]

(1,-5) and (6,-20)

To find the linear equation in slope-intercept form \( (y = mx + b) \) for the line passing through the points \((1, -5)\) and \((6, -20)\), we will first calculate the slope \( (m) \) using the formula:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Here, \((x_1, y_1) = (1, -5)\) and \((x_2, y_2) = (6, -20)\).

Substituting the values:

\[ m = \frac{-20 - (-5)}{6 - 1} = \frac{-20 + 5}{5} = \frac{-15}{5} = -3 \]

Now that we have the slope \( m = -3 \), we can use one of the points to find the y-intercept \( (b) \). We will use the point \((1, -5)\):

Using the equation:

\[ y = mx + b \]

Substituting \( x = 1 \), \( y = -5 \), and \( m = -3 \):

\[ -5 = -3(1) + b \]

This simplifies to:

\[ -5 = -3 + b \]

To solve for \( b \):

\[ b = -5 + 3 = -2 \]

Now that we have \( m = -3 \) and \( b = -2 \), we can write the slope-intercept form of the equation:

\[ y = -3x - 2 \]

In the required format, the final answer is:

\[ y=-3x-2 \]

(-4,-1) and (14,-37)

To find the linear equation in slope-intercept form \( (y = mx + b) \) for the line passing through the points \((-4, -1)\) and \((14, -37)\), we first calculate the slope \( (m) \) using the formula:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Here, \((x_1, y_1) = (-4, -1)\) and \((x_2, y_2) = (14, -37)\).

Substituting the values into the slope formula:

\[ m = \frac{-37 - (-1)}{14 - (-4)} = \frac{-37 + 1}{14 + 4} = \frac{-36}{18} = -2 \]

Now that we have the slope \( m = -2 \), we can use one of the points to find the y-intercept \( (b) \). We will use the point \((-4, -1)\):

Using the equation:

\[ y = mx + b \]

Substituting \( x = -4 \), \( y = -1 \), and \( m = -2 \):

\[ -1 = -2(-4) + b \]

This simplifies to:

\[ -1 = 8 + b \]

Solving for \( b \):

\[ b = -1 - 8 = -9 \]

Now we have \( m = -2 \) and \( b = -9 \), so we can write the slope-intercept form of the equation:

\[ y = -2x - 9 \]

In the required format, the final answer is:

\[ y=-2x-9 \]