Answer :
Certainly! Let's solve these questions step by step.
### Question 3: What is the \( y \)-intercept?
To find the \( y \)-intercept of a line that passes through the given points, we first need to determine the equation of the line in the slope-intercept form, \( y = mx + c \), where \( m \) is the slope and \( c \) is the \( y \)-intercept.
#### 1. Calculate the slope (\( m \))
The formula for the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Using the first two points \((-3, -8)\) and \((-1, 2)\):
[tex]\[ m = \frac{2 - (-8)}{-1 - (-3)} = \frac{2 + 8}{-1 + 3} = \frac{10}{2} = 5 \][/tex]
So, the slope \( m \) is \( 5 \).
#### 2. Calculate the \( y \)-intercept (\( c \))
We can use the slope-intercept form \( y = mx + c \) to find the \( y \)-intercept \( c \). Using one of the points, for example \((-3, -8)\), and substituting \( m = 5 \):
[tex]\[ -8 = 5(-3) + c \][/tex]
Solving for \( c \):
[tex]\[ -8 = -15 + c \][/tex]
[tex]\[ c = -8 + 15 \][/tex]
[tex]\[ c = 7 \][/tex]
Therefore, the \( y \)-intercept \( c \) is \( 7 \).
### Question 4: Write an equation in standard form.
The standard form of the equation of a line is \( Ax + By = C \). We already have the slope-intercept form as \( y = 5x + 7 \).
To convert this equation to standard form:
1. Move all terms to one side of the equation to set it equal to \( 0 \):
[tex]\[ y - 5x - 7 = 0 \][/tex]
2. Rearrange it to match \( Ax + By = C \):
[tex]\[ 5x - y = -7 \][/tex]
Thus, the equation in standard form is:
[tex]\[ 5x - y = -7 \][/tex]
### Summary
To answer the questions:
3. The \( y \)-intercept is \( 7 \).
4. The equation in standard form is [tex]\( 5x - y = -7 \)[/tex].
### Question 3: What is the \( y \)-intercept?
To find the \( y \)-intercept of a line that passes through the given points, we first need to determine the equation of the line in the slope-intercept form, \( y = mx + c \), where \( m \) is the slope and \( c \) is the \( y \)-intercept.
#### 1. Calculate the slope (\( m \))
The formula for the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Using the first two points \((-3, -8)\) and \((-1, 2)\):
[tex]\[ m = \frac{2 - (-8)}{-1 - (-3)} = \frac{2 + 8}{-1 + 3} = \frac{10}{2} = 5 \][/tex]
So, the slope \( m \) is \( 5 \).
#### 2. Calculate the \( y \)-intercept (\( c \))
We can use the slope-intercept form \( y = mx + c \) to find the \( y \)-intercept \( c \). Using one of the points, for example \((-3, -8)\), and substituting \( m = 5 \):
[tex]\[ -8 = 5(-3) + c \][/tex]
Solving for \( c \):
[tex]\[ -8 = -15 + c \][/tex]
[tex]\[ c = -8 + 15 \][/tex]
[tex]\[ c = 7 \][/tex]
Therefore, the \( y \)-intercept \( c \) is \( 7 \).
### Question 4: Write an equation in standard form.
The standard form of the equation of a line is \( Ax + By = C \). We already have the slope-intercept form as \( y = 5x + 7 \).
To convert this equation to standard form:
1. Move all terms to one side of the equation to set it equal to \( 0 \):
[tex]\[ y - 5x - 7 = 0 \][/tex]
2. Rearrange it to match \( Ax + By = C \):
[tex]\[ 5x - y = -7 \][/tex]
Thus, the equation in standard form is:
[tex]\[ 5x - y = -7 \][/tex]
### Summary
To answer the questions:
3. The \( y \)-intercept is \( 7 \).
4. The equation in standard form is [tex]\( 5x - y = -7 \)[/tex].
