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What is the equation of the line that passes through the points (−3,4) and (−6,3)

? Write the answer in slope-intercept form.(1 point)
y=
(
)x+

11 answers

To find the equation of the line that passes through the points \((-3, 4)\) and \((-6, 3)\), we will first calculate the slope \(m\) using the formula:

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

Let \((x_1, y_1) = (-3, 4)\) and \((x_2, y_2) = (-6, 3)\):

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

Now that we have the slope \(m = \frac{1}{3}\), we can use the point-slope form of the equation of a line:

\[ y - y_1 = m(x - x_1) \]

We'll use the point \((-3, 4)\):

\[ y - 4 = \frac{1}{3}(x + 3) \]

Now, distribute the slope on the right-hand side:

\[ y - 4 = \frac{1}{3}x + 1 \]

Now, add \(4\) to both sides to solve for \(y\):

\[ y = \frac{1}{3}x + 1 + 4 \] \[ y = \frac{1}{3}x + 5 \]

Thus, the equation of the line in slope-intercept form is:

\[ y = \frac{1}{3}x + 5 \]

What is the slope of the line y=5x+10
?(1 point)

The equation \( y = 5x + 10 \) is in slope-intercept form, which is given by the general equation \( y = mx + b \), where \( m \) represents the slope and \( b \) represents the y-intercept.

In this case, the slope \( m \) is \( 5 \).

Thus, the slope of the line is \( 5 \).

Interpret the equation y=−4x+10
. What type of function does this equation define? Describe its graph.(1 point)
Responses

This is a linear function. Its graph is a straight line with a slope of −4
and a y
-intercept of 10.
This is a linear function. Its graph is a straight line with a slope of negative 4 and a y -intercept of 10.

This is a linear function. Its graph is a straight line with a y
-intercept of −4
and a slope of 10.
This is a linear function. Its graph is a straight line with a y -intercept of negative 4 and a slope of 10.

This is a nonlinear function. Its graph is a curve.
This is a nonlinear function. Its graph is a curve.

This is a nonlinear function. Its graph has a maximum at (0,10)
.

The correct interpretation of the equation \( y = -4x + 10 \) is as follows:

This is a linear function. Its graph is a straight line with a slope of negative 4 and a y-intercept of 10.

Explanation:

  • Linear Function: The equation represents a linear function because it is in the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
  • Slope: The slope is \(-4\), which indicates that the line decreases (or goes downwards) as \( x \) increases.
  • Y-intercept: The y-intercept is \( 10 \), meaning the line crosses the y-axis at the point \((0, 10)\).

Graph Description:

  • The graph of this function will be a straight line that slopes downward from left to right, starting at the point \((0, 10)\) on the y-axis, and decreasing steeply due to the slope of \(-4\).
The equation of a function is y=2x−6
. If this function were graphed, what would its y
-intercept be?(1 point)
Responses

−3
negative 3

2
2

−6
negative 6

6
6

The equation of the function is given as \( y = 2x - 6 \).

In slope-intercept form \( y = mx + b \), the \( y \)-intercept is represented by \( b \).

In this case, the \( y \)-intercept \( b = -6 \).

Therefore, the correct response is:

−6 (negative 6).

Use the table to answer the question.

x
y

18 10
6 −3

Write the equation of the linear function represented by the table.

(1 point)
Responses

y=(1312)x−(192)
y equals left parenthesis Start Fraction 13 over 12 End Fraction right parenthesis x minus Start Fraction 19 over 2 End Fraction right parenthesis

y=(−1312)x+(72)
y equals left parenthesis Start Fraction negative 13 over 12 End Fraction right parenthesis x plus Start Fraction 7 over 2 End Fraction right parenthesis

y=(1213)x−(11113)
y equals left parenthesis Start Fraction 12 over 13 End Fraction right parenthesis x minus Start Fraction 111 over 13 End Fraction right parenthesis

y=−192
y equals Start Fraction negative 19 over 12 End Fraction
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To find the equation of the linear function represented by the table, we need to determine the slope \( m \) and the y-intercept \( b \).

Step 1: Determine the slope \( m \)

Using the points given in the table:

  • Point 1: \( (18, 10) \)
  • Point 2: \( (6, -3) \)

The formula for the slope \( m \) is:

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

Substituting in the values:

\[ m = \frac{-3 - 10}{6 - 18} = \frac{-13}{-12} = \frac{13}{12} \]

Step 2: Use the slope and one point to find the intercept

We can now use the slope \( m = \frac{13}{12} \) and one of the points (let's use \( (18, 10) \)) to find the y-intercept \( b \) using the point-slope form:

\[ y = mx + b \] \[ 10 = \left(\frac{13}{12}\right)(18) + b \]

Calculating \( \left(\frac{13}{12}\right)(18) \):

\[ 10 = \frac{13 \times 18}{12} + b \] \[ 10 = \frac{234}{12} + b \] \[ 10 = 19.5 + b \]

To isolate \( b \):

\[ b = 10 - 19.5 \] \[ b = -9.5 \] \[ b = -\frac{19}{2} \]

Step 3: Write the equation

Now we can write the equation of the line:

\[ y = \frac{13}{12}x - \frac{19}{2} \]

Conclusion

Thus, the correct response would be:

y = (13/12)x - (19/2).

What can you interpret about the equation y=−2x−8
?(1 point)
Responses

The slope is −2
, the y
-intercept is 8, and the line is falling from left to right.
The slope is negative 2 , the y -intercept is 8, and the line is falling from left to right.

The slope is −2
, the y
-intercept is −8
, and the line is falling from left to right.
The slope is negative 2 , the y -intercept is negative 8 , and the line is falling from left to right.

The slope is −2
, the y
-intercept is 8, and the line is rising from left to right.
The slope is negative 2 , the y -intercept is 8, and the line is rising from left to right.

The slope is 2, the y
-intercept is −8
, and the line is rising from left to right.

To interpret the equation \( y = -2x - 8 \):

  1. Identify the slope: The coefficient of \( x \) in the equation is \(-2\), which means the slope \( m \) is \(-2\).

  2. Identify the y-intercept: The constant term (the value when \( x = 0 \)) is \(-8\), so the y-intercept \( b \) is \(-8\).

  3. Determine the direction of the line: Since the slope is negative, the line falls from left to right.

Based on this analysis, the correct interpretation is:

The slope is −2, the y-intercept is −8, and the line is falling from left to right.