Derive the equation of the line in slope-intercept form. (1, 8) (0, 4)

Answer 1

To derive the equation of the line in slope-intercept form (y = mx + b) using the points (1, 8) and (0, 4), follow these steps:

1. Calculate the slope (m) using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Let's take the points as \( (x_1, y_1) = (0, 4) \) and \( (x_2, y_2) = (1, 8) \).

   Substituting the points into the formula: \[ m = \frac{8 - 4}{1 - 0} = \frac{4}{1} = 4 \]

2. Use the slope-intercept form of the equation, which is \( y = mx + b \). We know \( m = 4 \), so we can write: \[ y = 4x + b \]

3. Substitute one of the points to find b. You can use either point. Let's use the point (0, 4): \[ 4 = 4(0) + b \] This simplifies to: \[ b = 4 \]

4. Write the final equation: \[ y = 4x + 4 \]

Thus, the equation of the line in slope-intercept form is: \[ \boxed{y = 4x + 4} \]

Answer 2

same thing, but with (0, 2) (1, -1)

Answer 3

To derive the equation of the line in slope-intercept form (y = mx + b) using the points (0, 2) and (1, -1), follow these steps:

1. Calculate the slope (m) using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Let's take the points \( (x_1, y_1) = (0, 2) \) and \( (x_2, y_2) = (1, -1) \).

   Substituting the points into the formula: \[ m = \frac{-1 - 2}{1 - 0} = \frac{-3}{1} = -3 \]

2. Write the slope-intercept form of the equation, which is \( y = mx + b \): \[ y = -3x + b \]

3. Substitute one of the points to find b. We can use the point (0, 2): \[ 2 = -3(0) + b \] This simplifies to: \[ b = 2 \]

4. Write the final equation: \[ y = -3x + 2 \]

Thus, the equation of the line in slope-intercept form is: \[ \boxed{y = -3x + 2} \]

Answer 4

ok, now with (3, 0) and (0, -2)

Answer 5

To derive the equation of the line in slope-intercept form (y = mx + b) using the points (3, 0) and (0, -2), follow these steps:

1. Calculate the slope (m) using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] We'll take the points \( (x_1, y_1) = (0, -2) \) and \( (x_2, y_2) = (3, 0) \).

   Substituting the points into the formula: \[ m = \frac{0 - (-2)}{3 - 0} = \frac{0 + 2}{3} = \frac{2}{3} \]

2. Write the slope-intercept form of the equation, which is \( y = mx + b \): \[ y = \frac{2}{3}x + b \]

3. Substitute one of the points to find b. We can use the point (0, -2): \[ -2 = \frac{2}{3}(0) + b \] This simplifies to: \[ b = -2 \]

4. Write the final equation: \[ y = \frac{2}{3}x - 2 \]

Thus, the equation of the line in slope-intercept form is: \[ \boxed{y = \frac{2}{3}x - 2} \]

Answer 6

Jonas is saving money for an upcoming vacation. He currently has $50 in his savings account. He plans to add $3 each week. Write an equation that represents the relationship between the amount of money, y
y
, and the number of weeks, x
x
. Use the slope-intercept form y=mx+b
y
=
m
x
+
b
.(1 point)

Answer 7

To write an equation that represents the relationship between the amount of money \( y \) (in dollars) and the number of weeks \( x \), we can use the slope-intercept form, which is given by:

\[ y = mx + b \]

Here:

• \( m \) is the slope, which represents the amount added each week.
• \( b \) is the y-intercept, which represents the initial amount saved.

In this case:

• The initial amount \( b \) is $50, since that is how much Jonas currently has in his savings account.
• The amount he plans to add each week \( m \) is $3.

Now we can plug these values into the equation:

\[ y = 3x + 50 \]

Thus, the equation that represents the relationship between the amount of money \( y \) and the number of weeks \( x \) is:

\[ \boxed{y = 3x + 50} \]